Friday, September 25, 2015

calculus - Prove that xn=1+frac24+frac316+...+fracn4n1 converges



so i have got a sequence xn=1+24+316+464+...+n4n1
and i have to prove that it actually converges to some point, just by looking at it, it is clear to me that it does converge, if i would take its limit lim



as n increases the numerator becomes actually less than the
denominator, from that point it would converge, but how would i prove it.


Answer



Use comparison:

\sum_{k=0}^{\infty}{\frac{k}{4^{k-1}}}<\sum_{k=0}^{\infty}\frac{2^k}{4^{k-1}}=4\sum_{k=0}^{\infty}\bigg(\frac{2}{4}\bigg)^k


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