Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2018 / Paper 1 / 19 / b / ii

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- Solution ii

Solution

Rationalize the positive difference:

n^{4} + a^{2} - n^{2} = \frac{a ^{2}}{n ^{4} + a ^{2} + n ^{2}} .

Since the denominator is at least

2 n^{2}

,

0 < n^{4} + a^{2} - n^{2} \leq \frac{a ^{2}}{2 n ^{2}} .

The upper bound is a constant multiple of the convergent p-series with exponent two. The comparison test therefore proves

n = 1 \sum \infty (n^{4} + a^{2} - n^{2}) converges for every fixed a > 0.

The asymptotic equivalence of its summand to

a^{2} / (2 n^{2})

gives the same conclusion by limit comparison test.

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