Taylor Series for Cosine and Accuracy of Calculating Cosine 2

[stukbv]

Homework Statement

How many terms of the taylor series of the cosine function about c = 0 are needed to calculate cosine 2 to an accuracy of 1 / 10000

The Attempt at a Solution

I have said that |Rn(2)| = |cosⁿ⁺¹(a) 2ⁿ⁺¹/(n+1)!|<2ⁿ⁺¹/(n+1!)

Now i can't do it ...

 

[micromass]

Hi stukbv!

You have correctly calculated that the remainder must be smaller than $\frac{2^{n+1}}{(n+1)!}$

Now, the only thing you need to do is to see when

$$\frac{2^{n+1}}{(n+1)!}\leq \frac{1}{10000}$$

Perhaps calculate the left hand side for some values of n and see when the inequality occurs. Thus, calculate the left-hand side for n=1,2,3,...

 

[stukbv]

Wouldnt I need a calculator to do this - I am not allowed those in my exams...

 

[vela]

No, you don't. It may help to rewrite the inequality as

$$2^{n+1}\le \frac{(n+1)!}{10000}$$

 

[stukbv]

Sorry if I am being simple but I still don't see how this is meant to be a quick calculation like my exams seem to be implying?? Is there are link I am missing?

 

[vela]

As micromass said, just try increasing values of n until you find one for which the inequality holds. Your instructor probably assumes you have some familiarity with the first handful of powers of 2 and factorials. If not, it's not like it very long to calculate them.