Determine 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?

Re: 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}

Re: Determine whether the following series converges or diverges.

Quote:

Originally Posted by

**sjmiller** 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}

Im still not sure where to go from there. This is a geometric series, correct?

Re: Determine whether the following series converges or diverges.

Quote:

Originally Posted by

**sjmiller** 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}

Quote:

Originally Posted by

**Preston019** Im still not sure where to go from there. This is a geometric series, correct?

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.

Re: Determine whether the following series converges or diverges.

Quote:

Originally Posted by

**sjmiller** 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.

Okay, so is this true: ∑ ar^(-n-1) = a/(1-r)^-1

And how would I get ∑(-3/∏)^n into the proper form?

Re: Determine whether the following series converges or diverges.

Quote:

Originally Posted by

**Preston019** Okay, so is this true: ∑ ar^(-n-1) = a/(1-r)^-1

And how would I get ∑(-3/∏)^n into the proper form?

No that does not follow from the original formula.

Note that (-3/pi)^n can be written as (-3/pi)(-3/pi)^(n-1) where a = (-3/pi) and r = (-3/pi) and |r| < 1 so it indeed converges.

Re: Determine whether the following series converges or diverges.

Quote:

Originally Posted by

**sjmiller** No that does not follow from the original formula.

Note that (-3/pi)^n can be written as (-3/pi)(-3/pi)^(n-1) where a = (-3/pi) and r = (-3/pi) and |r| < 1 so it indeed converges.

So, then how do I put ∑(e/9)^(n+1) into the proper form?

Re: Determine whether the following series converges or diverges.

Re: Determine whether the following series converges or diverges.

Quote:

Originally Posted by

**Preston019** So, then how do I put ∑(e/9)^(n+1) into the proper form?

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.