Calculus #6: ∑ n = 1 4 n cos ⁡ ( n ) ( x ) \sum _{ n=1 }^{ 4n }{ \cos ^{ (n) } \left( x \right) }

Calculus Level 3

You may want to solve this problem first.

∑ n = 1 4 m cos ⁡ ( n ) ( x ) \large \sum _{ n=1 }^{ 4m }{ \cos ^{(n)} ( x) }

Find the sum above (to 2 decimal places), where m m is an integer.

Notation: f ( n ) ( x ) f^{(n)}(x) denotes to the n n th derrivative of the function f ( x ) f (x) .

See more calculus problems .


The answer is 0.

Ron Lauterbach by Ron Lauterbach , 18, Germany

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

Chew-Seong Cheong
Oct 25, 2017

∑ n = 1 4 m cos ⁡ ( n ) x = − sin ⁡ x − cos ⁡ x + sin ⁡ x + cos ⁡ x ⏟ = 0 − sin ⁡ x − cos ⁡ x + ⋯ − sin ⁡ x − cos ⁡ x + sin ⁡ x + cos ⁡ x ⏟ = 0 = 0 \begin{aligned} \sum_{n=1}^{4m} \cos^{(n)} x & = \underbrace{{\color{#3D99F6}- \sin x}{\color{#D61F06} - \cos x} {\color{#3D99F6} + \sin x}{\color{#D61F06} + \cos x}}_{=0} {\color{#3D99F6}- \sin x}{\color{#D61F06} - \cos x} + \cdots \underbrace{{\color{#3D99F6}- \sin x}{\color{#D61F06} - \cos x} {\color{#3D99F6} + \sin x}{\color{#D61F06} + \cos x}}_{=0} = \boxed{0} \end{aligned}

We note that the sum of every consecutive 4 terms is 0, therefore, the sum of 4 m 4m terms is also 0 \boxed{0} .

2 Helpful 0 Interesting 0 Brilliant 0 Confused

0 pending reports

×

Problem Loading...

Note Loading...

Set Loading...