Thursday, 14 April 2016

real analysis - How to prove this sequence is convergent?






Suppose that the series k=1ak converges. Prove that
lim





I tried to use the definition of convergence of \sum a_k but I'm struggling to find N such that given \epsilon >0, if n>N, then -\epsilon<\frac{1}{n}\sum_{k=1}^{n}ka_k<\epsilon but I don't know how to connect them



hope somebody can help


Answer




Let s_n=a_1+\cdots+a_n\to a. Then
\sum_{k=1}^n ka_k=\sum_{k=1}^n k(s_k-s_{k-1})=\sum_{k=1}^n ks_k-\sum_{k=1}^{n-1}(k+1)s_k=ns_n-\sum_{k=1}^{n-1}s_k.
Hence
\frac{1}{n}\sum_{k=1}^n ka_k=s_n-\frac{n-1}{n}\cdot\frac{1}{n-1}\sum_{k=1}^{n-1}s_k\to a-a=0.
We have used the fact that: If b_n\to b, then so does \,\,\dfrac{b_1+\cdots+b_n}{n}.




Note. However, it is not in general true that na_n\to 0.


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