Year 12s: what will be the first way you research your future options? >>

Year 12s: what will be the first way you research your future options? >>

Year 12s: what will be the first way you research your future options? >>

Faster way to do expansion of e^cos(x) (FP2/Edexcel)

username2796236

hello all

I am trying to find a more efficient way of expanding e^cos(x) up to the term in x^4. I did this by working out derivatives up to the fourth and then using f(0).

However this method was tedious and long winded. Is there a more efficient way of doing this problem ?

Many thanks

Original post by sideshowbobya

However this method was tedious and long winded. Is there a more efficient way of doing this problem ?

Many thanks

I don't know about efficient, but you know

e^x = 1 + x + x^2/2 + x^3/6 x^4/24 + o(x^5)

and so

e^{\cos x} = 1 + \cos x + \cos^2 x/2 + \cos^3 x /6 + \cos^4 x / 24 + o(\cos^5 x)

.

Now, you also know the expansion of

\cos x = 1-x^2/2 + x^4/24 + o(x^6)

, so... can you take it from here?

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