Exercise 7.4.8

Question

For each $n \in \mathbf{N}$, let
$$
h_{n}(x)=\left\{\begin{array}{ll}
1 / 2^{n} & 	ext { if } 1 / 2^{n}<x \leq 1 \
0 & 	ext { if } 0 \leq x \leq 1 / 2^{n}
\end{array},ight.
$$
and set $H(x)=\sum_{n=1}^{\infty} h_{n}(x)$. Show $H$ is integrable and compute $\int_{0}^{1} H$.

Ulisse Mini · Accepted Answer

$\int_0^1 h_i = (1/2)^n - (1/4)^n$. $H_n(x) = \sum^n_{i=1}h_i(x)$ is integrable, and $(H_n) \to H$ uniformly, so an application of the geometric series formula and the Integrable Limit Theorem gives us $\int^1_0 H = 2/3$.

Exercise 7.4.8

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Exercise 7.4.8

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