Determine whether the following series converges or diverges

∞

∑ [(-3/∏)^n + (e/9)^(n+1)]

n=1

If it converges, find its sum.

I don't know how to go about solving this one, can someone explain the steps that need to be taken?

- Sep 30th 2012, 12:27 PMPreston019Determine whether the following series converges or diverges.
Determine whether the following series converges or diverges

∞

∑ [(-3/∏)^n + (e/9)^(n+1)]

n=1

If it converges, find its sum.

I don't know how to go about solving this one, can someone explain the steps that need to be taken?

- Sep 30th 2012, 12:31 PMsjmillerRe: Determine whether the following series converges or diverges.
Distribute the sum over both terms, and if the series converges, both of those terms should converge.

If ∑a_{n }and ∑b_{n}are convergent series, then so are the series ∑ca_{n}(where c is a constant)

Then ∑(a_{n}+ b_{n}) = ∑a_{n + }∑b_{n} - Sep 30th 2012, 01:00 PMPreston019Re: Determine whether the following series converges or diverges.
- Sep 30th 2012, 01:20 PMsjmillerRe: Determine whether the following series converges or diverges.
Correct they are both geometric series. Remember the sum of such a series ∑ ar

^{n-1}= a/(1-r) provided that |r|<1

You need to get them into a form where you can easily determine the values of a and r. If |r| > 1 then the series diverges. - Sep 30th 2012, 02:23 PMPreston019Re: Determine whether the following series converges or diverges.
- Sep 30th 2012, 02:50 PMsjmillerRe: Determine whether the following series converges or diverges.
- Sep 30th 2012, 03:13 PMPreston019Re: Determine whether the following series converges or diverges.
- Sep 30th 2012, 03:18 PMPlatoRe: Determine whether the following series converges or diverges.
Lets do the using partial sums.

If $\displaystyle S_n = \sum\limits_{k = 1}^n {\left[ {\left( {\frac{{ - 3}}{\pi }} \right)^k + \left( {\frac{e}{9}} \right)^k } \right]} $ would you agree that $\displaystyle S_n=U_n+V_n$ where $\displaystyle U_n = \sum\limits_{k = 1}^n {\left[ {\left( {\frac{{ - 3}}{\pi }} \right)^k } \right]} $ and $\displaystyle V_n = \sum\limits_{k = 1}^n { {\left( {\frac{e}{9}} \right)^k } } $

Does each of $\displaystyle U_n~\&~V_n$ converge? If so, the sum of their limits is the sum of the original series. - Sep 30th 2012, 03:24 PMsjmillerRe: Determine whether the following series converges or diverges.
Same idea as the prior, instead of decreasing n by 1 we now want to decrease n by 2.

(e/9)^(n-1) = (e/9)^2(e/9)^(n-1)

Not that if you add the exponents together you get 2 + (n - 1) = n + 1

This is always your goal with a geometric series. Always check that the absolute value of r is less than 1, because if r is greater than 1 the series diverges.