(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Define I(x)= I( x - x_n ) =

{ 0 , when x < x_n

{ 1, when x >= x_n.

Let f be the monotone function on [0,1] defined by

[tex] f(x) = \sum_{n=1}^{\infty} \frac{1}{2^n} I ( x - x_n) [/tex]

where [tex] x_n = \frac {n}{n+1} , n \in \mathbb{N} [/tex].

Find [tex] \int_0^1 f(x) dx [/tex].

Leave your answer in the form of an infinite series.

2. Relevant equations

3. The attempt at a solution

I know the theorem, if f is monotone on [0,1], then f is Riemann integrable on [0,1]. I am also familiar with what the graph of this function looks like.

My calculation of what is under the function is,

[tex] \sum_{n=1}^{\infty} \frac{1}{2^n} \frac{1}{n^2 +3n +2} [/tex].

I need some advice for completing the problem (assuming that sum is correct).

Thanks

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# Trying to integrate a summation of a unit step function.

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