Do Sequences and Series Always Converge with Positive Decreasing Terms?

[SPhy]

Just wanted to know if I'm approaching this problem correctly.

1. The Problem

A. If the terms of a sequence of all positive terms go to zero, then the sequence must converge? True or false. Provide an example.

B.If the terms of a series of all positive terms go to zero, then the series converges? True or false. Provide an example.

2. Attempt

A. True, consider the sequence ace sub n, starting at n = 0 and going to infinity, where ace sub n is 1/(n^4+2)

The terms of this sequence decrease to zero and the sequence converges and the limit is 0.

B. False, consider the Harmonic series 1/n, the limit as n--->inf = 0, but the terms do not decrease to 0.

 

[LCKurtz]

Just wanted to know if I'm approaching this problem correctly.

1. The Problem

A. If the terms of a sequence of all positive terms go to zero, then the sequence must converge? True or false. Provide an example.

B.If the terms of a series of all positive terms go to zero, then the series converges? True or false. Provide an example.

2. Attempt

A. True, consider the sequence ace sub n, starting at n = 0 and going to infinity, where ace sub n is 1/(n^4+2)

The terms of this sequence decrease to zero and the sequence converges and the limit is 0.

B. False, consider the Harmonic series 1/n, the limit as n--->inf = 0, but the terms do not decrease to 0.

Please restate B more completely and carefully and be sure you answer the question. It is asking about a series.