# Science:Math Exam Resources/Courses/MATH105/April 2012/Question 08 (a)

MATH105 April 2012
Other MATH105 Exams

### Question 08 (a)

Short-answer questions. No credit will be given for the answer (even if it is correct) without the accompanying work.

The Maclaurin series for arctan x is given by

$\arctan x=\sum _{n=0}^{\infty }(-1)^{n}{\frac {x^{2n+1}}{2n+1}}$ which has radius of convergence equal to 1. Use this fact to compute the exact value of the series below:

$\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n+1)3^{n}}}$ (Hint: Note that $3^{n}=({\sqrt {3}})^{2n}$ ).

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