Question and Answers Forum

All Questions      Topic List

Others Questions

Previous in All Question      Next in All Question      

Previous in Others      Next in Others      

Question Number 149378 by fotosy2k last updated on 05/Aug/21

Answered by Ar Brandon last updated on 05/Aug/21

$$S=\psi \left(4\right)-\psi \left(1\right)=\frac{1}{3}+\frac{1}{2}+1+\psi \left(1\right)-\psi \left(1\right)=\frac{11}{6}$$$$S=(1-\cancel{\frac{1}{4}})+(\frac{1}{2}-\cancel{\frac{1}{5}})+(\frac{1}{3}-\cancel{\frac{1}{6}})+(\cancel{\frac{1}{4}}-\cancel{\frac{1}{7}})+\cdot \cdot \cdot +$$$$=1+\frac{1}{2}+\frac{1}{3}=\frac{11}{6}$$

Answered by gsk2684 last updated on 13/Sep/21

$$(1-\frac{1}{4})+(\frac{1}{2}-\frac{1}{5})+(\frac{1}{3}-\frac{1}{6})+$$$$(\frac{1}{4}-\frac{1}{7})+(\frac{1}{5}-\frac{1}{8})+(\frac{1}{6}-\frac{1}{9})+$$$$(\frac{1}{7}-\frac{1}{10})+(\frac{1}{8}-\frac{1}{11})+(\frac{1}{9}-\frac{1}{12})+...$$$$=1+\frac{1}{2}+\frac{1}{3}$$

Answered by mathmax by abdo last updated on 05/Aug/21

$${\mathrm{S}}_{\mathrm{n}}=\sum _{\mathrm{k}=1}^{\mathrm{n}}(\frac{1}{\mathrm{k}}-\frac{1}{\mathrm{k}+3})\Rightarrow {\mathrm{S}}_{\mathrm{n}}=\sum _{\mathrm{k}=1}^{\mathrm{n}}\frac{1}{\mathrm{k}}-\sum _{\mathrm{k}=1}^{\mathrm{n}}\frac{1}{\mathrm{k}+3}$$$$\sum _{\mathrm{k}=1}^{\mathrm{n}}\frac{1}{\mathrm{k}}={\mathrm{H}}_{\mathrm{n}},\sum _{\mathrm{k}=1}^{\mathrm{n}}\frac{1}{\mathrm{k}+3}=\sum _{\mathrm{k}=4}^{\mathrm{n}+1}\frac{1}{\mathrm{k}}$$$$=\sum _{\mathrm{k}=1}^{\mathrm{n}}\frac{1}{\mathrm{k}}+\frac{1}{\mathrm{n}+1}-1-\frac{1}{2}-\frac{1}{3}={\mathrm{H}}_{\mathrm{n}}+\frac{1}{\mathrm{n}+1}-\frac{3}{2}-\frac{1}{3}={\mathrm{H}}_{\mathrm{n}}+\frac{1}{\mathrm{n}+1}-\frac{11}{6}$$$$\Rightarrow {\mathrm{S}}_{\mathrm{n}}={\mathrm{H}}_{\mathrm{n}}-{\mathrm{H}}_{\mathrm{n}}-\frac{1}{\mathrm{n}+1}+\frac{11}{6}=\frac{11}{6}-\frac{1}{\mathrm{n}+1}\Rightarrow$$$$\sum _{\mathrm{n}=1}^{\mathrm{\infty }}(\frac{1}{\mathrm{n}}-\frac{1}{\mathrm{n}+3})=\frac{11}{6}$$

Commented by fotosy2k last updated on 05/Aug/21

$$thankyou$$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com