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Determine the Taylor series for the function
f(x)=∫01sinxxdx centered at x=0.f(x)=\int _0 ^1 \dfrac{\sin x}{x}dx ~\text{ centered at }~ x=0.f(x)=∫01xsinxdx centered at x=0.
Evaluate π22!−π44!+π66!−π88!+⋯ .\frac{\pi^2}{2!}-\frac{\pi^4}{4!}+\frac{\pi^6}{6!}-\frac{\pi^8}{8!}+\cdots.2!π2−4!π4+6!π6−8!π8+⋯.
f(x)=sinxcosx centered at x=0.f(x)=\sin x\cos x ~\text{ centered at }~ x=0.f(x)=sinxcosx centered at x=0.
Evaluate
∑n=1∞(−1)n+1n⋅2n. \sum_{n=1}^\infty \frac{(-1)^{n+1}}{n \cdot 2^n }. n=1∑∞n⋅2n(−1)n+1.
If f(x)=ex2f(x)=e^{x^2}f(x)=ex2, what is f(2016)(0)f^{(2016)}(0)f(2016)(0) ?
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