Power series for arctan

Question

Find the power series representation for

f(x)=∫x0 (arctan(t)/t)dt.

f(x)=∑n=1∞((−1)^en)(an)(x^pn)

where en=

A. n - 1

B. n

C. 0

and an=

and pn=

Answer 1

We know series tan^-1t = x - t³/3 + t⁵/5 -t⁷/7 + t⁹/9 - ...;

tan^-1t/t = 1 - t²/3 + t⁴/5 - t⁶/7 + t⁸/9 - . . .

∫₀^xtan^-1t/tdt = (t - t³/9 + t⁵/25 - t⁷/49 + t⁹/81 - . . .)₀^x = x - x³/9 + x⁵/25 - x⁷/49 + x⁹/81 - ... =

∑₀∞(-1)ⁿx^{2n+ 1}/(2n + 1)