Evaluate ((Σ_(k=1) ^(10) (∫_0 ^k (4u+1)du))/(5^2 Σ_(n=1) ^∞ (1/2)(Σ_(n=2) ^∞ (2/(m^2 +2m)))^(n−1) ))∫_(sin^(−1) (((−(√2))/2))) ^((π/2)cos(π/2)) (((1−secθsinθ)/((tanθ+cotθ)/(ϱ^θ −ϱ^(πi) ))))dθ


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Question Number 216886 by Engr_Jidda last updated on 23/Feb/25

$E v a l u a t e \frac{\underset{k = 1}{\sum^{10}} (\int_{0}^{k} (4 u + 1) d u)}{5^{2} \underset{n = 1}{\sum^{\infty}} \frac{1}{2} {(\underset{n = 2}{\sum^{\infty}} \frac{2}{m^{2} + 2 m})}^{n - 1}} \int_{{s i n}^{- 1} (\frac{- \sqrt{2}}{2})}^{\frac{π}{2} c o s \frac{π}{2}} (\frac{1 - s e c θ s i n θ}{\frac{t a n θ + c o t θ}{ϱ^{θ} - ϱ^{π i}}}) d θ$
Commented by MathematicalUser2357 last updated on 24/Feb/25

$w e c a n u s e t h e f o r m u l a t o c l e a n t h i s g i b b e r i s h$
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