Question 2.2: Let (xn) and (yn) be sequences in R. We want to show that if |xn| ≤ yn for all n ∈ N and ∑∞n=1 yn converges then ∑∞n=1 xn converges as well. For this we suggest the following proof: We have that lim yn = 0 and that |xn| ≤ yn for all n ∈ N. n→∞ Therefore the sequence (|xn|) converges to zero. The latter implies that∑∞n=1 |xn| converges and thus the series ∑∞n=1 xn converges.

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Sequence and Series

A sequence is defined as a collection of things. Series is defined to sum the things one by one in the sequence.  It was invented by German mathematician Carl Friedrich Gauss.