Verified answer

"An infinite geometric series will converge if the associated geometric sequence has a limit of 0."

This is true.

If the associated geometric sequence has a limit of 0, then either:

(a) all the terms are 0, or

(b) the common ratio, r, is such that |r| < 1.

In either case, the infinite geometric series will converge.

"All infinite arithmetic series diverge."

This is false.

Consider 0 + 0 + 0 + ..., with first term and common difference both equal to 0, which converges to 0.

"An infinite sum is equal to the limit of the associated sequence."

This is false.

Consider 1 + 1/2 + 1/4 + 1/8 + ... , where the limit of the sequence is 0 but the infinite sum is equal to 2.

Or consider 1 + 1/2 + 1/3 + 1/4 + ... , where the limit of the sequence is 0 but the infinite sum is not defined. (The series diverges.)

"Suppose an is an arithmetic sequence with d > 0. Then the sum of the series a1+a2+a3+...a12 must be positive."

This is false.

Consider a1 = -12 and d = 1, so that a12 = -1. All the terms are negative, so the sum is negative.