The integral is cos((8/9)x^4) from 0 to 1, I need to express this as an infinite series. I am having trouble finding the series any help is much appreciated.
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Start with cos t = Σ(n = 0 to ∞) (-1)^n t^(2n)/(2n)!.
Let t = (8/9)x^4:
cos((8/9)x^4) = Σ(n = 0 to ∞) (-1)^n ((8/9)x^4)^(2n)/(2n)!
......................= Σ(n = 0 to ∞) (-1)^n (8/9)^(2n) x^(8n)/(2n)!
Integrating term by term for x = 0 to 1 yields
∫(x = 0 to 1) cos((8/9)x^4) dx
= Σ(n = 0 to ∞) (-1)^n (8/9)^(2n) x^(8n+1)/[(8n+1) * (2n)!] {for x = 0 to 1}
= Σ(n = 0 to ∞) (-1)^n (8/9)^(2n)/[(8n+1) * (2n)!].
I hope this helps!
Let t = (8/9)x^4:
cos((8/9)x^4) = Σ(n = 0 to ∞) (-1)^n ((8/9)x^4)^(2n)/(2n)!
......................= Σ(n = 0 to ∞) (-1)^n (8/9)^(2n) x^(8n)/(2n)!
Integrating term by term for x = 0 to 1 yields
∫(x = 0 to 1) cos((8/9)x^4) dx
= Σ(n = 0 to ∞) (-1)^n (8/9)^(2n) x^(8n+1)/[(8n+1) * (2n)!] {for x = 0 to 1}
= Σ(n = 0 to ∞) (-1)^n (8/9)^(2n)/[(8n+1) * (2n)!].
I hope this helps!