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Exercise 3.10

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(a)

Let

P (n) = \frac{k}{n}

for

n \in ℕ

. By principles of probability,

\sum_{n \in ℕ} P (n)

must equal

1

\sum_{n \in ℕ} P (n) = k \sum_{n \in ℕ} \frac{1}{n} .

The sum on the right side of the equality is a divergent, harmonic series. Hence. the aforementioned principle of probably is violated. Contradiction.

(b)

\sum_{n \in ℕ} \frac{1}{n^{2}} = \frac{π^{2}}{6}

. Thus, letting

k

equal

\frac{6}{π^{2}}

, the principle of probability is satisfied.

fifthist

2021-12-05 00:00

Exercise 3.10

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