Question and Answers Forum

All Questions      Topic List

Relation and Functions Questions

Previous in All Question      Next in All Question      

Previous in Relation and Functions      Next in Relation and Functions      

Question Number 103763 by mathmax by abdo last updated on 17/Jul/20

calculate{ Σ_(n=0) ^∞ (−1)^n x^n }×{Σ_(n=o) ^∞ (x^(2n) /(n+1))}

$$\mathrm{calculate}\left\{\:\sum_{\mathrm{n}=\mathrm{0}} ^{\infty} \left(−\mathrm{1}\right)^{\mathrm{n}} \mathrm{x}^{\mathrm{n}} \right\}×\left\{\sum_{\mathrm{n}=\mathrm{o}} ^{\infty} \:\frac{\mathrm{x}^{\mathrm{2n}} }{\mathrm{n}+\mathrm{1}}\right\} \\ $$

Answered by mathmax by abdo last updated on 18/Jul/20

here we use Σ a_n ×Σ b_n =Σ c_n c_n =Σ_(i+j =n) a_i b_j =Σ_(i=0) ^n a_i b_(n−i) =Σ_(i=0) ^n (−1)^i x^i (x^(2(n−i)) /(n−i +1)) =Σ_(i=0) ^n (((−1)^i )/(n−i+1)) x^(i+2n−2i) =Σ_(i=0) ^n (((−1)^i )/(n−i+1))x^(2n−i) ⇒ Σ_(n=0) ^∞ (−1)^n x^n ×Σ_(n=0) ^∞ (x^(2n) /(n+1)) =Σ_(n=0) ^∞ Σ_(i=0) ^n (((−1)^i )/(n−i+1)) x^(2n−i)

$$\mathrm{here}\:\mathrm{we}\:\mathrm{use}\:\Sigma\:\mathrm{a}_{\mathrm{n}} ×\Sigma\:\mathrm{b}_{\mathrm{n}} =\Sigma\:\mathrm{c}_{\mathrm{n}} \: \\ $$$$\mathrm{c}_{\mathrm{n}} =\sum_{\mathrm{i}+\mathrm{j}\:=\mathrm{n}} \:\mathrm{a}_{\mathrm{i}} \mathrm{b}_{\mathrm{j}} \:=\sum_{\mathrm{i}=\mathrm{0}} ^{\mathrm{n}} \:\mathrm{a}_{\mathrm{i}} \mathrm{b}_{\mathrm{n}−\mathrm{i}} \\ $$$$=\sum_{\mathrm{i}=\mathrm{0}} ^{\mathrm{n}} \:\left(−\mathrm{1}\right)^{\mathrm{i}} \:\mathrm{x}^{\mathrm{i}} \:\:\frac{\mathrm{x}^{\mathrm{2}\left(\mathrm{n}−\mathrm{i}\right)} }{\mathrm{n}−\mathrm{i}\:+\mathrm{1}}\:=\sum_{\mathrm{i}=\mathrm{0}} ^{\mathrm{n}} \:\frac{\left(−\mathrm{1}\right)^{\mathrm{i}} }{\mathrm{n}−\mathrm{i}+\mathrm{1}}\:\mathrm{x}^{\mathrm{i}+\mathrm{2n}−\mathrm{2i}} \\ $$$$=\sum_{\mathrm{i}=\mathrm{0}} ^{\mathrm{n}} \:\frac{\left(−\mathrm{1}\right)^{\mathrm{i}} }{\mathrm{n}−\mathrm{i}+\mathrm{1}}\mathrm{x}^{\mathrm{2n}−\mathrm{i}} \:\Rightarrow \\ $$$$\sum_{\mathrm{n}=\mathrm{0}} ^{\infty} \left(−\mathrm{1}\right)^{\mathrm{n}} \:\mathrm{x}^{\mathrm{n}} \:×\sum_{\mathrm{n}=\mathrm{0}} ^{\infty} \:\frac{\mathrm{x}^{\mathrm{2n}} }{\mathrm{n}+\mathrm{1}}\:=\sum_{\mathrm{n}=\mathrm{0}} ^{\infty} \sum_{\mathrm{i}=\mathrm{0}} ^{\mathrm{n}} \:\frac{\left(−\mathrm{1}\right)^{\mathrm{i}} }{\mathrm{n}−\mathrm{i}+\mathrm{1}}\:\mathrm{x}^{\mathrm{2n}−\mathrm{i}} \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com