# Summing Taylor Series: Tips & Tricks

• Cpt Qwark
In summary, recognizing patterns and using the fact that sin(x) is an odd function can help in rewriting the Taylor/Maclaurin series for sin(x) in summation notation.
Cpt Qwark
Expanding the series to the $$n^{th}$$ derivative isn't so hard, however I'm having trouble with the summation. Any tips for the summation?
e.g. taylor series for $$sinx$$ around x=0 in summation notation is $$\sum^\infty_{n=0} \frac{x^{4n}}{2n!}$$
Thanks.

Cpt Qwark said:
Expanding the series to the $$n^{th}$$ derivative isn't so hard, however I'm having trouble with the summation. Any tips for the summation?
e.g. taylor series for $$sinx$$ around x=0 in summation notation is $$\sum^\infty_{n=0} \frac{x^{4n}}{2n!}$$
Thanks.
No, it isn't. For one thing, sin(x) is an odd function while your series includes only even power of x. The Taylor's series for sin(x) about x= 0 is $$\sum_{n=o}^\infty \frac{(-1)^n x^{2n+1}}{(2n+1)!}$$. What you have appears to be the Taylor's series, about x= 0, for $cos(x^2)$, except that the denominator should be (2n)! rather than 2n!.

In any case, what do you mean "having trouble with the summation". What are you trying to do?

No, the Taylor series sum around x=0 (i.e. the Maclaurin series sum) for ## \sin x ## is $$\sum_{k=0}^{\infty} \frac{(-1)^k x^{(1+2 k)}}{(1+2 k)!}$$. How did you get to the expression you wrote?

Last edited:
HallsofIvy said:
No, it isn't. For one thing, sin(x) is an odd function while your series includes only even power of x. The Taylor's series for sin(x) about x= 0 is $$\sum_{n=o}^\infty \frac{(-1)^n x^{2n+1}}{(2n+1)!}$$. What you have appears to be the Taylor's series, about x= 0, for $cos(x^2)$, except that the denominator should be (2n)! rather than 2n!.

In any case, what do you mean "having trouble with the summation". What are you trying to do?

MrAnchovy said:

No, the Taylor series sum around x=0 (i.e. the Maclaurin series sum) for ## \sin x ## is $$\sum_{k=0}^{\infty} \frac{(-1)^k x^{(1+2 k)}}{(1+2 k)!}$$. How did you get to the expression you wrote?

Yeah sorry turns out it was mistook for another expression.
Anywas, what I meant was I had trouble rewriting the taylor/maclaurin series with a summation notation (Σ). Are there supposed to be patterns that you're supposed to recognise (such as the negative sign for sine and cosine functions) or something?

I'm not always in favour of Khan Academy but this might help.

Or is it just getting from ## x - \frac{x^3}{3!} +\frac{x^5}{5!} -\frac{x^7}{7!} +\frac{x^9}{9!} - ... ## to the summation formula that is giving you problems?

If so then yes, you need to practice recognising parts of terms like this:
• first note you can always write ## x ## as ## \frac{x^1}{1!} ##
• now notice you have odd numbers 1, 3, 5, 7, 9...: you can generate these with ## 2k + 1 ## - that gives you ## \frac{x^{2k+1}}{(2k+1)!} ##
• now you just need the alternating + and - signs: -1 to an even power is 1 and to an odd power is -1 so, making sure you start off with the right one (you want the 0th term to have ## 1 = (-1)^0 ## not ## -1 = (-1)^{0+1} ##) you have ## (-1)^k ##
• put them all together, add the sum remembering to go from ## k=0 ## - full marks!

## What is a Taylor Series?

A Taylor Series is a mathematical representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point.

## Why is it useful to sum Taylor Series?

Summing Taylor Series allows for an approximation of a function, which can be useful for solving complex problems in physics, engineering, and other sciences.

## What are some tips for summing Taylor Series?

Some tips for summing Taylor Series include using properties of derivatives, understanding the convergence of the series, and choosing the appropriate number of terms for the desired level of accuracy.

## What are some common mistakes when summing Taylor Series?

Some common mistakes when summing Taylor Series include using an incorrect number of terms, neglecting to consider the convergence of the series, and making errors in calculating the derivatives of the function.

## How can I apply Taylor Series in my research?

Taylor Series can be applied in a variety of fields, such as physics, engineering, economics, and statistics. It can be used to approximate functions, solve differential equations, and analyze data.

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