Answer to Question #348806 in Calculus for Sarita bartwal

Question #348806

Check whether the sequence (an), where



an = 1/ (n+1) + 1/(n+2) +....+1/(2n) is convergent or not


1
Expert's answer
2022-06-08T01:40:23-0400

Consider


1+12+...+1n+1n+1+...+12n1+\dfrac{1}{2}+...+\dfrac{1}{n}+\dfrac{1}{n+1}+...+\dfrac{1}{2n}=(1+12+...+1n)+(1n+1+...+12n)=(1+\dfrac{1}{2}+...+\dfrac{1}{n})+(\dfrac{1}{n+1}+...+\dfrac{1}{2n})

Then



1n+1+...+12n=(1+12+13+...+1n\dfrac{1}{n+1}+...+\dfrac{1}{2n}=(1+\dfrac{1}{2}+\dfrac{1}{3}+...+\dfrac{1}{n}+1n+1+...+12n1+12n)2(12+14+...+12n)+\dfrac{1}{n+1}+...+\dfrac{1}{2n-1}+\dfrac{1}{2n})-2(\dfrac{1}{2}+\dfrac{1}{4}+...+\dfrac{1}{2n})=112+1314+...+12n112n=1-\dfrac{1}{2}+\dfrac{1}{3}-\dfrac{1}{4}+...+\dfrac{1}{2n-1}-\dfrac{1}{2n}=k=12n(1)k+1k=\displaystyle\sum_{k=1}^{2n}\dfrac{(-1)^{k+1}}{k}

Consider ln(1+x)\ln(1+x) for x(1,x]x\in(-1, x]



ln(1+x)=k=1(1)k+1xkk\ln(1+x)=\displaystyle\sum_{k=1}^{\infin}\dfrac{(-1)^{k+1}x^k}{k}

Then for x=1x=1



ln2=ln(1+1)=k=1(1)k+1(1)kk\ln2=\ln(1+1)=\displaystyle\sum_{k=1}^{\infin}\dfrac{(-1)^{k+1}(1)^k}{k}=k=1(1)k+1k=\displaystyle\sum_{k=1}^{\infin}\dfrac{(-1)^{k+1}}{k}

We see that



limnan=limn(1n+1+...+12n)\lim\limits_{n\to\infin}a_n=\lim\limits_{n\to\infin}(\dfrac{1}{n+1}+...+\dfrac{1}{2n})=limnk=12n(1)k+1k=k=1(1)k+1k=ln2=\lim\limits_{n\to\infin}\displaystyle\sum_{k=1}^{2n}\dfrac{(-1)^{k+1}}{k}=\displaystyle\sum_{k=1}^{\infin}\dfrac{(-1)^{k+1}}{k}=\ln2

Therefore the sequence (an)(a_n) is convergent.


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