Taylor Series Applications

[Anewk]

What is the general procedure for using Taylor Series to evaluate:

i) sums

eg.\(\displaystyle \sum_{n=4}^{\infty }\frac{n(n-1)2^n}{3^n}\)

ii) limits

eg. \(\displaystyle \lim_{x\rightarrow 2}\frac{x^2-4}{ln(x-1)}\)

iii) derivatives

eg. Find \(\displaystyle f^{(11)}(0)\) of \(\displaystyle f(x)=x^3sin(x^2)\)

iv) integrals

eg. \(\displaystyle \int_{0}^{1} \frac{1}{2-x^3}dx\)

 

[jvicens]

Hi there!

The general procedure for using Taylor Series to evaluate sums, limits, derivatives, and integrals is as follows:

i) For sums, we start by writing out the general form of the Taylor Series expansion for the given function. In this case, the function is \frac{n(n-1)2^n}{3^n}. Then, we plug in the given range of values for n (in this case, n=4 to infinity) into the Taylor Series expansion. This will give us an infinite series, which we can then simplify to a finite sum by using algebraic manipulation or by taking a certain number of terms.

ii) For limits, we start by writing out the general form of the Taylor Series expansion for the given function. In this case, the function is \frac{x^2-4}{ln(x-1)}. Then, we take the limit of the Taylor Series expansion as x approaches the given value (in this case, x=2). This will give us the value of the limit.

iii) For derivatives, we start by writing out the general form of the Taylor Series expansion for the given function. In this case, the function is f(x)=x^3sin(x^2). Then, we take the nth derivative of the Taylor Series expansion and evaluate it at the given point (in this case, x=0). This will give us the value of the nth derivative of the function at that point.

iv) For integrals, we start by writing out the general form of the Taylor Series expansion for the given function. In this case, the function is \frac{1}{2-x^3}. Then, we integrate the Taylor Series expansion from the lower limit (in this case, x=0) to the upper limit (in this case, x=1). This will give us the value of the integral.

I hope this helps! Let me know if you have any further questions.