Infinite Derivation

Calculus Level 1

lim ⁡ q → ∞ ∑ n = 1 4 q d n d x n sin ⁡ ( x ) = ? \large \lim_{q \to \infty} \sum_{n=1}^{4q}\frac{d^n}{dx^n}\sin(x) = ? where q ∈ N q \in \mathbb N .


The answer is 0.

Mohd. Hamza by Mohd. Hamza , 18, India

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1 solution

Mohd. Hamza
Oct 11, 2018

Just expand the summation and you are done!

lim ⁡ q → ∞ ∑ n = 1 4 q d n d x n sin ⁡ ( x ) \large\lim_{q\to\infty}\sum_{n=1}^{4q}\frac{d^n}{dx^n}\sin(x) = d d x sin ⁡ ( x ) \frac{d}{dx}\sin(x) + d 2 d x 2 sin ⁡ ( x ) \frac{d^2}{dx^2}\sin(x) + d 3 d x 3 sin ⁡ ( x ) \frac{d^3}{dx^3}\sin(x) + d 4 d x 4 sin ⁡ ( x ) \frac{d^4}{dx^4}\sin(x) +.....

= cos ⁡ ( x ) \cos(x) +( − sin ⁡ ( x ) -\sin(x) )+( − cos ⁡ ( x ) -\cos(x) )+ sin ⁡ ( x ) \sin(x) +...

=0

1 Helpful 0 Interesting 0 Brilliant 0 Confused

= cos ⁡ ( x ) \cos(x) +( − sin ⁡ ( x ) -\sin(x) )+( − cos ⁡ ( x ) -\cos(x) )+ sin ⁡ ( x ) \sin(x) +...

Note that this expression does not converge unless the number of terms in this expression is a multiple of 4.

Pi Han Goh - 2 years, 7 months ago

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