 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2004/paper-1/2/iv/solution

For

n \geq 1

, the ratio of positive terms is

\frac{( n + 1 ) ^{4} e ^{- (n + 1)^{2}}}{n ^{4} e ^{- n^{2}}} = (1 + \frac{1}{n})^{4} e^{- 2 n - 1} ⟶ 0.

The ratio test proves the series converges. Its zero term at

n = 0

has no effect.

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