Question and Answers Forum

All Questions      Topic List

Differentiation Questions

Previous in All Question      Next in All Question      

Previous in Differentiation      Next in Differentiation      

Question Number 158691 by mnjuly1970 last updated on 07/Nov/21

Σ_(n=1) ^∞ (( (−1)^( n) H_( 2n) )/(2n)) =?

$$ \\ $$$$\:\:\:\:\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\:\left(−\mathrm{1}\right)^{\:{n}} \:\mathcal{H}_{\:\mathrm{2}{n}} }{\mathrm{2}{n}}\:=? \\ $$

Answered by qaz last updated on 08/Nov/21

Σ_(n=1) ^∞ (((−1)^n )/(2n))H_(2n) =Σ_(n=1) ^∞ (−1)^(n−1) ∫_0 ^1 x^(2n−1) ln(1−x)dx =∫_0 ^1 ((ln(1−x))/(1+x^2 ))dx =∫_0 ^(π/4) ln(1−tan x)dx =∫_0 ^(π/4) ln(cos x−sin x)−lncos xdx =∫_0 ^(π/4) ln((√2)sin ((π/4)−x))−lncos xdx =(π/8)ln2+∫_0 ^(π/4) lntan xdx =(π/8)ln2−G

$$\underset{\mathrm{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\left(−\mathrm{1}\right)^{\mathrm{n}} }{\mathrm{2n}}\mathrm{H}_{\mathrm{2n}} \\ $$$$=\underset{\mathrm{n}=\mathrm{1}} {\overset{\infty} {\sum}}\left(−\mathrm{1}\right)^{\mathrm{n}−\mathrm{1}} \int_{\mathrm{0}} ^{\mathrm{1}} \mathrm{x}^{\mathrm{2n}−\mathrm{1}} \mathrm{ln}\left(\mathrm{1}−\mathrm{x}\right)\mathrm{dx} \\ $$$$=\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{ln}\left(\mathrm{1}−\mathrm{x}\right)}{\mathrm{1}+\mathrm{x}^{\mathrm{2}} }\mathrm{dx} \\ $$$$=\int_{\mathrm{0}} ^{\pi/\mathrm{4}} \mathrm{ln}\left(\mathrm{1}−\mathrm{tan}\:\mathrm{x}\right)\mathrm{dx} \\ $$$$=\int_{\mathrm{0}} ^{\pi/\mathrm{4}} \mathrm{ln}\left(\mathrm{cos}\:\mathrm{x}−\mathrm{sin}\:\mathrm{x}\right)−\mathrm{lncos}\:\mathrm{xdx} \\ $$$$=\int_{\mathrm{0}} ^{\pi/\mathrm{4}} \mathrm{ln}\left(\sqrt{\mathrm{2}}\mathrm{sin}\:\left(\frac{\pi}{\mathrm{4}}−\mathrm{x}\right)\right)−\mathrm{lncos}\:\mathrm{xdx} \\ $$$$=\frac{\pi}{\mathrm{8}}\mathrm{ln2}+\int_{\mathrm{0}} ^{\pi/\mathrm{4}} \mathrm{lntan}\:\mathrm{xdx} \\ $$$$=\frac{\pi}{\mathrm{8}}\mathrm{ln2}−\mathrm{G} \\ $$

Commented by mnjuly1970 last updated on 08/Nov/21

very nice sir qaz

$${very}\:{nice}\:{sir}\:{qaz} \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com