A question from university entrance examination in Turkey (LYS-2015)

Calculus Level 3

Let f n : [ n π , ( n + 1 ) π ] → R f_{n}:\left[ n\pi ,\left( n+1\right) \pi \right] \rightarrow\mathbb{R} be defined f n ( x ) = 1 5 n ∣ sin ⁡ x ∣ f_{n}\left( x\right) =\dfrac{1}{5^{n}} |\sin x| for all n ∈ N n\in \mathbb{N} .

What is the sum of the areas between all f n f_{n} 's and x x -axis?

3/2 5/2 9/5 7/5 8/5

G Gürsoy by G Gürsoy , 36, Turkey

This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try refreshing the page, (b) enabling javascript if it is disabled on your browser and, finally, (c) loading the non-javascript version of this page . We're sorry about the hassle.

1 solution

Kellen Atkins
Aug 13, 2018

We can write this as ∑ n = 0 ∞ ( ∫ n π ( n + 1 ) π ∣ sin ⁡ ( x ) ∣ ⋅ 1 5 n d x ) ∑ n = 0 ∞ 1 5 n ( ∫ n π ( n + 1 ) π ∣ sin ⁡ ( x ) ∣ d x ) ∑ n = 0 ∞ 1 5 n ( ∫ 0 π sin ⁡ ( x ) d x ) ∑ n = 0 ∞ 1 5 n ( 2 ) = 2 1 − 1 5 = 5 2 \sum_{n=0}^\infty \left(\int_{n\pi}^{\left(n+1\right)\pi}\left|\sin\left(x\right)\right|\cdot\frac{1}{5^n}dx\right) \\ \sum_{n=0}^\infty\frac{1}{5^n}\left(\int_{n\pi}^{\left(n+1\right)\pi}\left|\sin\left(x\right)\right|dx\right) \\ \sum_{n=0}^\infty\frac{1}{5^n}\left(\int_0^{\pi}\sin\left(x\right)dx\right) \\ \sum_{n=0}^\infty\frac{1}{5^n}\left(2\right)=\frac{2}{1-\frac{1}{5}}= \frac{5}{2 }

2 Helpful 0 Interesting 0 Brilliant 0 Confused

2 pending reports

Vote up reports you agree with

×

Problem Loading...

Note Loading...

Set Loading...