How should I prove convergence using Cauchy sequences \sum_{k=1}^n\frac{\sin(k^3+1)}{(4k+1)(4k+5)}

Nettie Potts

Nettie Potts

Answered question

2022-02-26

How should I prove convergence using Cauchy sequences
k=1nsin(k3+1)(4k+1)(4k+5)

Answer & Explanation

vazen2bl

vazen2bl

Beginner2022-02-27Added 9 answers

You have
|xn+pxn|=|k=n+1n+psin(k2+1)(4k+1)(4k+5)|
k=n+1n+p|sin(k2+1)(4k+1)(4k+5)|
=14k=n+1n+p(1(4k+1)1(4k+5))
=14(14n+514(n+p)+5)
14(4n+5)
and so choosing N>116ϵ will provide an N for you to use in proving that [xn]1 is Cauchy.

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