Question 10 - A-Level Maths

Exam Board	OCR
Module	Further Pure Core 1 (Further Pure Core 1)
Year	2018
Session	September
Topic	Taylor series
Type	Direct substitution into standard series

Using the Maclaurin series for $\ln ( 1 + x )$, find the first four terms in the series expansion for $\ln \left( 1 + 3 x ^ { 2 } \right)$.
Find the range of $x$ for which the expansion is valid.
Find the exact value of the series $$\frac { 3 ^ { 1 } } { 2 \times 2 ^ { 2 } } - \frac { 3 ^ { 2 } } { 3 \times 2 ^ { 4 } } + \frac { 3 ^ { 3 } } { 4 \times 2 ^ { 6 } } - \frac { 3 ^ { 4 } } { 5 \times 2 ^ { 8 } } + \ldots .$$