# calculus

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The sum from 0 to infinity of (-1)^n(3/3^n) is convergent or divergent? If convergent, what is the sum?

I got that it's convergent and the sum is 9/2, but that's wrong.

• calculus -

It is not clear without sufficient parentheses what the expression really is.
I assume it to be:
Sum((-1)^n * (3/3^n)) for n=0 -> ∞
This is an alternating geometric series.

(9/2) is the correct sum for the geometric series (non-alternating).

Write out the first few terms of the series:
3 - 3/3 + 3/9 - 3/27 + 3/81 - ...
which can be regrouped into two geometric series:
3(1+1/9+1/81+...) - (1+1/9+1/81+...)
=3(9/8) - (9/8)
=2(9/8)
=9/4

Note that if the minus sign becomes a plus sign, we get the geometric sum of 9/2.

• calculus -

Oh, right . I forgot all about the -1. Thanks.

• calculus -

You're welcome!

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