Do Sequences and Series Always Converge with Positive Decreasing Terms?

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SUMMARY

The discussion centers on the convergence of sequences and series with positive decreasing terms. It is established that a sequence of positive terms converges if its terms approach zero, as demonstrated by the sequence defined by an = 1/(n4 + 2), which converges to 0. Conversely, a series of positive terms does not necessarily converge just because its terms approach zero, exemplified by the Harmonic series 1/n, which diverges despite its terms tending to zero.

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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.
 
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SPhy said:
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.
 

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