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Show that, if $\sum a_{k} x^{k}$ and $\sum b_{k} x^{k}$ both converge to the same sum on some interval, then $a_{k}=b_{k}$ for each $k$...

Question

Show that, if $\sum a_{k} x^{k}$ and $\sum b_{k} x^{k}$ both converge to the same sum on some interval, then $a_{k}=b_{k}$ for each $k$

Show that, if $\sum a_{k} x^{k}$ and $\sum b_{k} x^{k}$ both converge to the same sum on some interval, then $a_{k}=b_{k}$ for each $k$



Answers

(a) Find examples to show that if $\sum a_{k}$ converges, then $\sum a_{k}^{2}$ may diverge or converge. (b) Find examples to show that if $\sum a_{k}^{2}$ converges, then $\quad \sum a_{k}$ may diverge or converge.


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