Past exam of the short options paper shared by Prelims and Parts A and B of the physics course of the University of Oxford / 2016 / S1 / 1 / 3

Table of contents
- Solution 3

Solution

Separate the

n = 0

term and pair

n

with

- n

. For small

α

,

\frac{( - 1 ) ^{n}}{( n - α ) ^{2}} + \frac{( - 1 ) ^{n}}{( n + α ) ^{2}} = \frac{2 ( - 1 ) ^{n}}{n ^{2}} + \frac{6 α ^{2} ( - 1 ) ^{n}}{n ^{4}} + O (α^{4} / n^{6}) .

Compare the constant and quadratic coefficients with the supplied expansion after subtracting

α^{- 2}

. It follows that

n \geq 1 \sum \frac{( - 1 ) ^{n}}{n ^{2}} = - \frac{π ^{2}}{12}, n \geq 1 \sum \frac{( - 1 ) ^{n}}{n ^{4}} = - \frac{7 π ^{4}}{720} .

The paired expansion is uniformly summable on any sufficiently small closed interval of

α

, so coefficient comparison is legitimate.

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