MHB Convergence of Series with Square Roots

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Determine whether the following series converge or diverge. You may use any appropriate test provided you explain your answer.

$$\sum^{\infty}_{n = 1} (\frac{1}{\sqrt{n}} - \frac{1}{\sqrt{n + 1}})$$

So it looks like this problem wants me to do telescoping series or something of that nature but I really want to avoid writing out those terms, can't I just split this into two sums and say this:

$$\sum^{\infty}_{n = 1} \frac{1}{\sqrt{n}} - \sum^{\infty}_{n = 1}\frac{1}{\sqrt{n + 1}}$$

Evaluating both of the series using p series test since, p $$\le$$ 1 for both series then the whole series diverges?

Is this correct?
 
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shamieh said:
Determine whether the following series converge or diverge. You may use any appropriate test provided you explain your answer.

$$\sum^{\infty}_{n = 1} (\frac{1}{\sqrt{n}} - \frac{1}{\sqrt{n + 1}})$$

So it looks like this problem wants me to do telescoping series or something of that nature but I really want to avoid writing out those terms,

If I were you, I would try to curb your distaste for writing out those terms. ;)

can't I just split this into two sums and say this:

$$\sum^{\infty}_{n = 1} \frac{1}{\sqrt{n}} - \sum^{\infty}_{n = 1}\frac{1}{\sqrt{n + 1}}$$

Evaluating both of the series using p series test since, p $$\le$$ 1 for both series then the whole series diverges?

Is this correct?

No, because you run into $\infty-\infty$ issues (that subtraction is undefined).
 
So I can't use p series at all? Are you sure about that?
 
shamieh said:
So I can't use p series at all? Are you sure about that?

Pretty sure! And I do think the original series converges, so the p series test simply doesn't apply in this situation.
 
So would you suggest I write this out as a telescoping series and see what I'm left with?
 
shamieh said:
Determine whether the following series converge or diverge. You may use any appropriate test provided you explain your answer.

$$\sum^{\infty}_{n = 1} (\frac{1}{\sqrt{n}} - \frac{1}{\sqrt{n + 1}})$$

So it looks like this problem wants me to do telescoping series or something of that nature but I really want to avoid writing out those terms, can't I just split this into two sums and say this:

$$\sum^{\infty}_{n = 1} \frac{1}{\sqrt{n}} - \sum^{\infty}_{n = 1}\frac{1}{\sqrt{n + 1}}$$

Evaluating both of the series using p series test since, p $$\le$$ 1 for both series then the whole series diverges?

Is this correct?

I'd write down what the finite sum (up to N terms) would be and then seeing what happens as $\displaystyle \begin{align*} N \to \infty \end{align*}$

$\displaystyle \begin{align*} \sum_{n = 1}^N{ \left( \frac{1}{\sqrt{n}} - \frac{1}{\sqrt{n+1}} \right) } &= \left( \frac{1}{\sqrt{1}} - \frac{1}{\sqrt{2}} \right) + \left( \frac{1}{\sqrt{2}} - \frac{1}{\sqrt{3}} \right) + \left( \frac{1}{\sqrt{3}} - \frac{1}{\sqrt{4}} \right) + \dots + \left( \frac{1}{\sqrt{N}} - \frac{1}{\sqrt{N + 1} } \right) \\ &= \frac{1}{\sqrt{1}} - \frac{1}{\sqrt{N + 1}} \\ &= 1 - \frac{1}{\sqrt{N + 1}} \end{align*}$

and so

$\displaystyle \begin{align*} \sum_{n = 1}^{\infty}{ \left( \frac{1}{\sqrt{n}} - \frac{1}{\sqrt{n + 1}} \right) } &= \lim_{N \to \infty}\sum_{n = 1}^N{ \left( \frac{1}{\sqrt{n}} - \frac{1}{\sqrt{n + 1}} \right) } \\ &= \lim_{N \to \infty} \left( 1 - \frac{1}{\sqrt{N + 1}} \right) \\ &= 1 - 0 \\ &= 1 \end{align*}$

The sum is clearly convergent (and actually can be evaulated).

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shamieh said:
Actually, no, the series diverges. So how are you getting converges? Looks like the p series will indeed work then?

Look.
https://www.wolframalpha.com/input/?i=summation++++1/sqrt(n)+-+1/(sqrt(n)+++1)

It would help if you write out the series correctly. You have put $\displaystyle \begin{align*} \sum_{ n = 1}^{\infty}{ \left( \frac{1}{\sqrt{n}} - \frac{1}{\sqrt{n} + 1} \right) } \end{align*}$ when it should be $\displaystyle \begin{align*} \sum_{n = 1}^{\infty}{ \left( \frac{1}{\sqrt{n}} - \frac{1}{\sqrt{n + 1}} \right) } \end{align*}$. The series is convergent.
 
It would help if you write out the series correctly. You have put $\displaystyle \begin{align*} \sum_{ n = 1}^{\infty}{ \left( \frac{1}{\sqrt{n}} - \frac{1}{\sqrt{n} + 1} \right) } \end{align*}$ when it should be $\displaystyle \begin{align*} \sum_{n = 1}^{\infty}{ \left( \frac{1}{\sqrt{n}} - \frac{1}{\sqrt{n + 1}} \right) } \end{align*}$. The series is convergent.
Yes that would help now wouldn't it? :rolleyes:
 
Thank you for typing that out, I found that same result. Do I need to say that the series converges to 1 or just simply that it is convergent?
 
It depends on what the question is asking. If it asks if the series is convergent, just say it's convergent. If it asks what the sum converges to, say it.
 

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