Nettie Potts

2022-02-26

How should I prove convergence using Cauchy sequences

$\sum _{k=1}^{n}\frac{\mathrm{sin}({k}^{3}+1)}{(4k+1)(4k+5)}$

vazen2bl

Beginner2022-02-27Added 9 answers

You have

$|{x}_{n+p}-{x}_{n}|=\left|\sum _{k=n+1}^{n+p}\frac{\mathrm{sin}({k}^{2}+1)}{(4k+1)(4k+5)}\right|$

$\le \sum _{k=n+1}^{n+p}\left|\frac{\mathrm{sin}({k}^{2}+1)}{(4k+1)(4k+5)}\right|$

$=\frac{1}{4}\sum _{k=n+1}^{n+p}(\frac{1}{(4k+1)}-\frac{1}{(4k+5)})$

$=\frac{1}{4}(\frac{1}{4n+5}-\frac{1}{4(n+p)+5})$

$\le \frac{1}{4(4n+5)}$

and so choosing$N>\frac{1}{16\u03f5}$ will provide an N for you to use in proving that $\left[{x}_{n}\right]}_{1}^{\mathrm{\infty}$ is Cauchy.

and so choosing

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