Question

test the series for convergence or divergence using the alternating series test

the sum from n=1 to infinity of (-1)^n/(3n+1)

I said it converges, is this true?

Answer 1

Hi:

You are correct. The test criteria are,

1) The terms a_n are of decreasing sequence; clearly 1/(3n+1) is decreasing on [1, inf].

2) Limit(n->inf)[a_n] = 0.

Both criteria having been met, we conclude convergence.

Regards,

Rich B.

Answer 2

"If the sequence An converges to 0, and each An is smaller than An-1 (i.e. the sequence An is monotone decreasing), then the series converges."

In this case, An = 1 / (3n + 1)

The limit as n->infinity (An) = 0

A1 = 1 / 4
A2 = 1 / 7

Therefore, by the Alternating Series Test the series converges.