f[n_Integer] := Nest[-D[#, x]/2 &, Cos[x], n]
Sum[f[n], {n, 0, Infinity}](*does not work*)
The sum of that series can be manually found as
$$s(x) = \frac{4}{5} \cos x + \frac{2}{5} \sin x$$
Is ...
Recall that the harmonic series $\displaystyle{\sum_{n=1}^{\infty}\frac{1}{n}}$ diverges. This is because we may bound the partial sums below, like so: $$1+\frac{1}{2}+\left(\frac{1}{3}+\frac{1}{4}\right)+\left(\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8}\right)+\cdots >$$ $$\frac{1}{2}+\frac{1}{2}+\left(\frac{1}{4}+\frac{1}{4}\right)+\left(\frac{1}{8}+\frac{1}{8}+\frac{1}{8}+\frac{1}{8}\right…
Well, perhaps it's not really my favourite book, but it's certainly right up there with the most heavily thumbed tomes on my office bookshelf. I'm referring to Tables of Integrals, Series and Products , by Gradshteyn and Ryzhik. I picked up a used copy of the 4 th ed. (1965) for about $5 some years ago at Powell's bookstore in Portland, and it's saved me more anguish and time than I can possibly …
