Taking the integral of a series

[Guidenable]

Homework Statement

Find the exact sum of:

\sum 2/(n7ⁿ)

n=1->∞

The Attempt at a Solution

Let S_n denote the nth partial sum.

ln(S_n) = \Sigma ln(2/(n7ⁿ))

=\Sigma ln2 - lnn - nln7

= nln2 - (ln1 + ln7 + ln2 + 2ln7 + ln3 + 3ln7 ...)
= nln2 - ln(n!) - ln7\Sigma n
= ln (2ⁿ)/n! - ln7 (\Sigma n)

I'm not sure where to go from here, or if I'm even going in the right direction.

 

[jbunniii]

Do you know how these two power series relate to each other?
\sum z^{n-1}
\sum \frac{1}{n} z^n

 

[Guidenable]

No, but now I intend to find out. A hint would be appreciated though.

 

[Guidenable]

Never mind. Those two series are related by a factor of 1/zn. I should have looked a bit closer :/.

 

[jbunniii]

Hint: \int z^{n-1} dz = \frac{1}{n} z^n

 

[vela]

Homework Statement

Find the exact sum of:

\sum 2/(n7ⁿ)

n=1->∞

The Attempt at a Solution

Let S_n denote the nth partial sum.

ln(S_n) = \Sigma ln(2/(n7ⁿ))

=\Sigma ln2 - lnn - nln7

= nln2 - (ln1 + ln7 + ln2 + 2ln7 + ln3 + 3ln7 ...)
= nln2 - ln(n!) - ln7\Sigma n
= ln (2ⁿ)/n! - ln7 (\Sigma n)

I'm not sure where to go from here, or if I'm even going in the right direction.

This approach won't work because the very first step is wrong. It's not true that ln(a+b) = ln a + ln b.

 

[Ray Vickson]

Homework Statement

Find the exact sum of:

\sum 2/(n7ⁿ)

n=1->∞

The Attempt at a Solution

Let S_n denote the nth partial sum.

ln(S_n) = \Sigma ln(2/(n7ⁿ))

=\Sigma ln2 - lnn - nln7

= nln2 - (ln1 + ln7 + ln2 + 2ln7 + ln3 + 3ln7 ...)
= nln2 - ln(n!) - ln7\Sigma n
= ln (2ⁿ)/n! - ln7 (\Sigma n)

I'm not sure where to go from here, or if I'm even going in the right direction.

Stop right there. What you have done is invalid: the log of a sum is not the sum of the logs. For example, if you claim that log(2 + 3) = log(2) + log(3) you would have log(2) + log(3) = log(2*3) = 6 (because the log of a product is the sum of the logs), so your method would give log(5) = log(6), hence 5 = 6.

RGV

 

[Guidenable]

Thanks for the correction, I normally know log rules, just that one slipped. You guys mean bringing the log into the sum, right?

 

[Guidenable]

Hint: \int z^{n-1} dz = \frac{1}{n} z^n

So I just evaluate the integral?

 

[jbunniii]

So I just evaluate the integral?

Yes, do you see how this can be applied to this problem?

 

[Ray Vickson]

Thanks for the correction, I normally know log rules, just that one slipped. You guys mean bringing the log into the sum, right?

I thought that was what I said.

RGV

 

[Guidenable]

Yes, do you see how this can be applied to this problem?

Yes, I understand now, thank you very much!

 

[Eats Dirt]

Yes, do you see how this can be applied to this problem?

Hey I was just wondering how you can take the integral of a series to find the sum? Don't series only include integers and will not add up to the same thing as a integral? For example my calculus textbook says when doing the integral test specifically states that the integral is not the sum of the series. Can someone please tell me what I am missing? Thank you.

 

[jbunniii]

Hey I was just wondering how you can take the integral of a series to find the sum? Don't series only include integers and will not add up to the same thing as a integral? For example my calculus textbook says when doing the integral test specifically states that the integral is not the sum of the series. Can someone please tell me what I am missing? Thank you.

If
f(z) = \sum a_n z^n
then, under certain conditions, you can perform manipulations like this:
\int f(z) dz = \int \sum a_n z^n dz = \sum \int a_n z^n dz = \sum \frac{a_n}{n+1} z^{n+1} dz
You need to provide justification for interchanging the order of integration and summation. Uniform convergence of the power series is a sufficient condition.

This kind of trick is useful for finding sums such as \sum (a_n / n) z^n, provided you have a closed-form expression for \sum a_n z^n. Term by term differentiation of the power series can also be useful in some situations.

 

[SammyS]

Hey I was just wondering how you can take the integral of a series to find the sum? Don't series only include integers and will not add up to the same thing as a integral? For example my calculus textbook says when doing the integral test specifically states that the integral is not the sum of the series. Can someone please tell me what I am missing? Thank you.

The idea here is to recognize that some fairly well-known power series in x (or z as jbunniii seems to like) is involved here, and that this power series represents a function of x. Then plug-in a value for x which gives the series of interest in this problem.