Problems

In Problems 1-10 find power series for f'(x) and for ∫₀^xf(t) dt.

In Problems 11-34 find a power series for the given function and determine its radius of convergence.

35            Check the formulas d(sinh x)/dx = cosh x, d(cosh x)/dx = sinh x by differentiating the power series.

36            Prove that if the power serieshas finite radius of convergence r, then the power series

has radius of convergence r/b (b > 0).

37            Prove that if has finite radius of convergence r, then

has radius of convergence √r.

38            Prove that if

have radii of convergence r and s respectively and r ≤ s, then f(x) + g(x) has a radius of convergence of at least r.

39            Show that if has radius of convergence r, then for any positive integer p,

has radius of convergence r.

40            Evaluate , using the derivative of the power series

41             Evaluate , using the first and second derivatives of