Question

the hundredth derivative sinx/x

Answer 1

\[
y={\sin x\over x}\\
y={1\over x}\left(x-{x^3\over 3!}+{x^5\over5!}-\ldots+(-1)^{n}{x^{2n+1}\over(2n+1)!}+\ldots \right)
\]

Answer 2

\[y=1-{x^2\over3!}+{x^4\over 5!}+\ldots +(-1)^n{x^{2n}\over(2n+1)!}\]

Answer 3

so, the hundredth derivative will start from
\(2n=100\)

Answer 4

\[y=\sum_{n=0}^\infty(-1)^n\frac{x^{2n}}{(2n+1)!}\\
y^{(k)}=\sum_{n=50}^\infty (-1)^n\frac{x^{2n-1}}{2n+1}
\]

Answer 5

this is sum of a geometric series to infinity...
get the first term, the comon ratio and bingo