Advanced-calculus-Q-If-0-1-0-1-xln-x-ln-2-y-1-xy-dxdy-n-1-1-n-3-n-1-2-Find-

Question Number 203964 by mnjuly1970 last updated on 03/Feb/24

A d v a n c e d c a l c u l u s \dots

Q : I f, \int_{0}^{1} \int_{0}^{1} \frac{x \ln (x) \ln^{2} (y)}{1 - x y} d x d y = λ \underset{n = 1}{\sum^{\infty}} \frac{1}{n^{3} {(n + 1)}^{2}}

\Rightarrow F i n d, λ = ?

Answered by witcher3 last updated on 03/Feb/24

nice Probleme

\forall x, y \in [0, 1 [\Rightarrow xy < 1

\frac{1}{1 - xy} = \underset{k = 0}{\sum^{n}} x^{k} y^{k}

\int_{0}^{1} \int_{0}^{1} \frac{xln (x) \ln^{2} (y)}{1 - xy} dxdy = \int_{0}^{1} \int_{0}^{1} \sum_{k ⩾ 0} x^{k + 1} y^{k} \ln (x) \ln^{2} (y) dy

= \sum_{k ⩾ 0} \int_{0}^{1} x^{k + 1} \ln (x) dx \int_{0}^{1} y^{k} \ln^{2} (y) dy

\sum_{k ⩾ 0} (- \frac{1}{{(k + 2)}^{2}}) . (\frac{2}{{(k + 1)}^{3}}); k + 1 = n; - 2 \sum_{n ⩾ 1} \frac{1}{n^{3} {(n + 1)}^{2}}

λ = - 2

Commented by mnjuly1970 last updated on 03/Feb/24

t h a n k s a l o t s i r w i c h e r

Commented by witcher3 last updated on 04/Feb/24

withe Pleasur

Answered by MathematicalUser2357 last updated on 06/Feb/24

λ = - 2