Let \(\displaystyle{k}\in{\mathbb{{N}}}\). Prove that \(\displaystyle{0}{ < }{\frac{{1}}{{k}}}-{\ln{{\left({1}+{\frac{{1}}{{k}}}\right)}}}{

ropowiec2gkc

ropowiec2gkc

Answered question

2022-03-24

Let kN. Prove that
0<1kln(1+1k)<12k2

Answer & Explanation

Cecilia Nolan

Cecilia Nolan

Beginner2022-03-25Added 13 answers

You can use the Taylor series for log(1+x) to get
ln(1+1k)=1k12k2+13k314k4+
Now apply the alternating series theorem which says if you truncate the alternating series the error is smaller than and of the same sign as the first neglected term.
Karsyn Wu

Karsyn Wu

Beginner2022-03-26Added 17 answers

HINT
By 1k=x we need to show that
0<xln(1+x)<12x2x12x2ln(1+x)x,x(0,1]

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