so i have got a sequence xn=1+24+316+464+...+n4n−1
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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