Pratica di Shadowing: Simple Guide to Series Convergence Tests - Impara a parlare inglese con YouTube

So we are going to talk about what convergence tests to use

⏸ In Pausa

Velocità:

249 frasi

Se le frasi sono troppo corte o troppo lunghe, clicca su Edit per modificarle.

So we are going to talk about what convergence tests to use

when we're trying to figure out whether an infinite series converges.

We'll go over when to use the test for divergence,

the geometric series, the p-series,

the comparison test, the limit comparison test,

the alternating series test, the integral test, and the ratio test.

All of that good stuff.

So I've broken down every infinite series we'll generally see on an exam into four different categories here,

and we're going to go over how to deal with each one.

Let's start out with the basic category.

This is talking about those series that we already know whether they converge or diverge right away.

And those two are going to be the geometric series,

the sum from n equals 1 to infinity of r to the n,

which we know converges when the absolute value of r is less than 1,

and the sum from n equals 1 to infinity of 1 over n to the power of p.

And that's going to converge when p is greater than 1.

So this is the geometric series and the p series formula.

When we see a series that we can write in terms of one of these two things,

we should always do that first because we can instantly get to an answer just like that.

But these aren't the most common types of series,

So let's take a look at some of the other categories.

The next one is almost basic series.

And these are series that are very,

very close to a geometric series or a p-series,

but are just not quite there.

There's a little something off.

For example, the sum from n equals 1 to infinity of 1 over the square root of n cubed plus 5.

So when n gets really,

really, really large, notice inside this square root,

n cubed is going to blow up.

It's going to be massive,

but 5 is just going to stay as 5.

So the bigger we get in this series,

the less this plus 5 is going to matter.

It's basically just going to be n cubed.

So we can write this as basically the sum from n

equals 1 to infinity of 1 over the square root of n cubed.

But notice that the square root of n cubed is something

that we can write as n to the power of 3 halves.

And if we have the sum from n equals 1 to infinity of 1 over n to the 3 halves,

that is a p series,

and we know that it converges.

So these kinds of series where it's basically a p series,

it's basically geometric, but there's a little thing that's throwing it off,

these are almost basic series.

series.

And the way that we evaluate this kind of series is always by using a comparison test.

So there are two comparison tests which are the standard comparison test,

or the direct comparison test,

and the limit comparison test.

And it doesn't generally matter which one we use,

the limit comparison test will usually work in more of the cases,

but sometimes we can get away with the comparison test as well.

So in the case of this series here,

notice on the bottom n cubed plus 5,

that's always going to be bigger than n cubed.

And if we have something bigger in the denominator,

that's going to make our results smaller.

So we know that 1 over the square root of n cubed plus 5,

that's always going to be less than 1 over the square root of n cubed.

Of course on an exam we'll have to prove this more rigorously,

but this is the general reasoning that we need.

Once we get to this,

we can use the direct comparison test to say,

well, the sum of 1 over the square root of n cubed,

that's a p series and that converges.

Because this is less, this must also converge.

So that's how we can use the comparison test or the limit comparison test to evaluate these almost basic series.

And we'll go through a few exercises at the end where we can practice identifying which type of series we're dealing with.

So the next category we're going to look at is alternating series.

And this is a very simple category.

It is exactly what it sounds like.

So if we have the sum from n equals 1 to infinity of negative 1 to the n times b sub n,

where b sub n is positive,

this is an alternating series.

And any time we have an alternating series,

it's always a good idea to try using the alternating series test to see whether it converges.

As an example,

if we had the sum from n equals 1 to infinity

of negative 1 to the n over the natural log of n plus 1,

this is an alternating series and we're going to try using the alternating series test on it.

In this case we can actually prove that this is an alternating series that converges by that test.

So there will be some cases where an alternating series,

we can't prove it converges by the alternating series test,

but it's always a good thing to try first if we can pull out a negative 1 to the n.

Now the last type of series we're going to talk about is the weird series.

And these are the series that don't really fall into any of the categories that we already talked about.

As an example,

we might have the sum from n equals 1 to infinity of n times e to the negative n squared.

So this clearly isn't one of our basic forms,

and it's not really almost basic either because not only do we have this n,

but it's not just e to the negative n,

it's e to the negative n squared and that's just strange,

it's not alternating either.

So this is a case where we're going to have to use a different method.

And there are two different methods that we use for weird series.

The first is the integral test.

And we use the integral test any time we can integrate the series.

So in this case, the sum from n equals one to infinity of n e to the negative n squared,

100

well we are able to do the integral from 1 to infinity of x e to the negative x squared dx.

101

And as long as the conditions for the integral test are satisfied,

102

meaning it's continuous, positive, and decreasing over time,

103

we are able to use the integral test for this kind of series.

104

There's one other kind of weird series,

105

which I'll give an example up here.

106

For example,

107

the sum from n equals 1 to infinity of n factorial

108

times 3 to the power of 2n over 4 to the n times n plus 3 factorial.

109

So in this case, the important thing about these kinds of weird series is we have powers and we have factorials.

110

So I'll write that here.

111

When we have powers and factorials,

112

And when I say powers,

113

I'm talking specifically about n in the power.

114

In that case, we're going to use the ratio test to evaluate these.

115

So in this case, if we use the ratio test for this series,

116

we would be able to evaluate it to a particular number

117

that we could use to prove that this either converges or diverges.

118

So those are the two kinds of weird series we're talking about.

119

If you can do the integral,

120

do the integral test, as long as the conditions are satisfied.

121

And if we have factorials and powers flying around everywhere,

122

that's when we're going to try the ratio test.

123

So now that we've identified all of these categories,

124

let's go through a few exercises where we try to figure out

125

which type of series we're looking at so we know which test to apply.

126

Let's start out with a basic one,

127

the sum from n equals 1 to infinity of 5 halves to the power of negative 2n.

128

This already sort of looks like a geometric series, right?

129

So we have something and it's sort of to the power of n.

130

So before we look at anything else,

131

let's see if we can interpret it as just this basic form of a geometric series.

132

To do that, we want to take a look at the exponent first.

133

because we know that if we have 5 halves to the negative 2n,

134

we can actually split up the powers as 5 halves to the power of negative 2

135

and then to the power of n.

136

And once we do this,

137

we can rewrite the series in a form that we like a lot more.

138

We'll have the sum from n equals 1 to infinity.

139

Well, what's 5 halves to the negative 2?

140

That's going to be the same thing as 2 fifths squared, right?

141

We have to flip it over for a negative power.

142

that's going to be 4 divided by 25, right?

143

2 squared over 5 squared,

144

and then we raise that to the power of n.

145

And now this is exactly in the form of a geometric series,

146

r to the n, and our r is 4 over 25.

147

So if we have our ratio being 4 25ths,

148

that's less than 1, which means we have proved that this series converges by the geometric series.

149

Next up,

150

let's take a look at the sum from n equals 1

151

to infinity of n squared times 3 to the n over n factorial.

152

Notice in this case, this is already looking kind of weird.

153

So we might be looking at this weird category to start out.

154

And we also notice that we have factorials and we have numbers raised to the power of n.

155

And those two things indicate to us that we should probably be using the ratio test.

156

So in the case of this series,

157

we won't go through all the work right now because we're just trying to identify which test to use,

158

but if we use the ratio test on this,

159

we would be able to prove that the common ratio as n approaches infinity is less than 1,

160

and therefore that this series converges by the ratio test.

161

The next one we're going to take a look at will be a little different.

162

It'll be the sum from n equals 1 to infinity of

163

n to the sixth plus n squared over n to the seventh plus 3.

164

Now, notice in this case,

165

as n gets really, really large,

166

n to the sixth is going to be much,

167

much, much bigger than n squared,

168

and n to the seventh is going to be much,

169

much bigger than 3,

170

which means

171

that really what we're looking at is very similar to the

172

sum from n equals 1 to infinity of n to the sixth over n to the seventh.

173

But we can write this,

174

n to the sixth over n to the seventh,

175

we put the powers together,

176

as just 1 over n.

177

So we can say that our series that we have here is very much like 1 over n,

178

as n gets very large.

179

And this is a type of p-series.

180

So in reality, the series that we're looking at here is an instance of an almost basic series.

181

So we're going to want to use either the comparison test

182

or the limit comparison test to turn what we have here into the p-series.

183

Now because we have this n squared on the top,

184

using the comparison test is going to be a little strange,

185

so I think it would be easier in this instance to use the limit comparison test.

186

we would use the limit comparison test to compare this series to the sum of 1 over n.

187

And then we would get the value of 1 as our answer

188

and because 1 is the result that we get and the sum of 1 over n diverges,

189

the harmonic series, p is less than or equal to 1,

190

then we know that our original series diverges as well.

191

Next up,

192

let's take a look at the sum from n equals 1

193

to infinity of negative 1 to the n over n factorial plus n squared plus 3.

194

Now you might be tempted to look at this as a weird series because of the n factorial,

195

and that might work, but also notice that we have a negative 1 to the n in the numerator here,

196

which means this is an alternating series.

197

the first thing we always want to try is the alternating series test.

198

And in this case, we would actually be able to prove that this sum converges by the alternating series test.

199

So we're going to look at one more series here,

200

which is the sum from n equals 1 to infinity of 1 over the inverse tangent of 6n.

201

Now this is not a basic series.

202

It's not almost basic either because we have this inverse trig function kind of strange.

203

It's not alternating and we might be tempted to look at this as a weird series,

204

but we should hold on first and examine what this series actually looks like.

205

If we think about the inverse tangent function,

206

what would the graph of that function look like?

207

Well, as n on this x-axis here increases,

208

the inverse tangent is going to look like this.

209

It's going to have a vertical asymptote at the value of pi over 2,

210

which means if we divide by the inverse tangent,

211

this is not going to go to 0.

212

So if we take the limit as n approaches infinity of the inside of the sum,

213

1 over the inverse tangent of 6n,

214

that result is not going to be zero.

215

And if this limit is not equal to zero,

216

then by the test for divergence, this series must diverge.

217

So this exercise is a quick reminder that any time we're doing series,

218

no matter what it looks like,

219

we always have to remember the test for divergence,

220

because it might seem that it would be a good idea to try some other series test,

221

but as long as we can prove that the inside of the sum never goes to zero.

222

We can get off really easy just doing a limit and proving that it diverges.

223

So, these are the four categories that it's always good to remember

224

when you're trying to figure out which convergence test to use.

225

If you see a sum that you can write in the form of sum number to the power of n,

226

that's a geometric series, so that's a basic form that you can evaluate directly.

227

The same goes for anything you can write as 1 over n to the power of p.

228

The next category is things that are almost these basic forms.

229

For example, the sum of 1 over the square root of n cubed plus 5.

230

If we could take out that 5,

231

then we would get the n to the p that we want.

232

So this is an almost basic sum,

233

and these are where we use the comparison

234

and the limit comparison to write this sum as we say it's very similar to another form

235

that we know how to evaluate.

236

The next category is alternating series.

237

Anytime a series is alternating,

238

the first thing we want to try is the alternating series test

239

to see if we can use that to prove that it converges.

240

And the final category is the weird series.

241

So they aren't basic, they aren't almost basic,

242

and they're not alternating either.

243

When we get these kinds of series,

244

depending on the form, we're either going to use the integral test if we can do the integral,

245

or we'll use the ratio test if it has things to the power of n and factorials.

246

And the last thing to remember is the test for divergence.

247

Anytime the series isn't going to go to zero,

248

we can just take that limit,

249

show it's not zero, and then we're done.

TRENDING

Popolari

Cos'è la tecnica dello Shadowing?

Shadowing è una tecnica di apprendimento delle lingue supportata da studi scientifici, originariamente sviluppata per la formazione dei traduttori professionisti e resa popolare dal poliglotta Dr. Alexander Arguelles. Il metodo è semplice ma potente: ascolti un audio in inglese di madrelingua e lo ripeti immediatamente ad alta voce — come un'ombra che segue il parlante con un ritardo di solo 1–2 secondi. A differenza dell'ascolto passivo o degli esercizi di grammatica, lo shadowing costringe il tuo cervello e i muscoli della bocca a elaborare e riprodurre simultaneamente i modelli di discorso reale. La ricerca dimostra che migliora significativamente la precisione della pronuncia, l'intonazione, il ritmo, il discorso connesso, la comprensione dell'ascolto e la fluidità del parlato — rendendolo uno dei metodi più efficaci per la preparazione alla prova di speaking dell'IELTS e per la comunicazione reale in inglese.

☕ Offrici un caffè

ShadowingEnglish rimane gratuito al 100% grazie al tuo supporto. I costi del server e IA sono elevati — il tuo caffè ci aiuta ad andare avanti! 🙏

Dona via PayPal

ShadowingEnglish.com – Pratica di Shadowing in Inglese

Parla inglese fluentemente usando la tecnica dello Shadowing. Ascolta video di YouTube con madrelingua, ripeti frase per frase e costruisci una pronuncia e fluidità reale — usata da studenti IELTS in tutto il mondo.

Pratica di Shadowing: Simple Guide to Series Convergence Tests - Impara a parlare inglese con YouTube

Popolari

Contesto & Sfondo

Le 5 Frasi Chiave per la Comunicazione Quotidiana

Guida Passo-Passo per Shadowing

Cos'è la tecnica dello Shadowing?

Pratica di Shadowing: Simple Guide to Series Convergence Tests - Impara a parlare inglese con YouTube

Scarica l'app

Contesto & Sfondo

Le 5 Frasi Chiave per la Comunicazione Quotidiana

Guida Passo-Passo per Shadowing

Cos'è la tecnica dello Shadowing?