- #1
hereiscassie
- 6
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Doing a MacLaurin Series and more!
The function f is defined by f(x) = 1/(1+x^3). The MacLaurin series for f is given by
1 - x^3 + x^6 - x^9 +...+ (-1)^n(x^3n) +...
which converges to f(x) for -1 < x < 1.
a) Find the first three nonzero terms and the general term for the MacLaurin series for f '(x).
b) Use your results from part (a) to find the sum of the infinite series -3/2^2 + 6/2^5 - 9/2^8 +...+ (-1)^n[3n/ 2^(3n - 1)] +...
c) Find the first four nonzero terms and the general term for the MacLaurin series representing the intergral from 0 to x of f(t) dt.
d) Use the first three nonzero terns of the infinite series found in part (c) to approxiamate the integral from 0 to .5 of f(t) dt. What are the properties of the terms of the series representing the integral from 0 to .5 of f(t) dt that guarantee that this approxiamation is within 1/10,000 of the exact value of the integral.
I don't even know what to do in this one! I need some major help! especially in part b and d! DX
Please and Thank You! :)
The function f is defined by f(x) = 1/(1+x^3). The MacLaurin series for f is given by
1 - x^3 + x^6 - x^9 +...+ (-1)^n(x^3n) +...
which converges to f(x) for -1 < x < 1.
a) Find the first three nonzero terms and the general term for the MacLaurin series for f '(x).
b) Use your results from part (a) to find the sum of the infinite series -3/2^2 + 6/2^5 - 9/2^8 +...+ (-1)^n[3n/ 2^(3n - 1)] +...
c) Find the first four nonzero terms and the general term for the MacLaurin series representing the intergral from 0 to x of f(t) dt.
d) Use the first three nonzero terns of the infinite series found in part (c) to approxiamate the integral from 0 to .5 of f(t) dt. What are the properties of the terms of the series representing the integral from 0 to .5 of f(t) dt that guarantee that this approxiamation is within 1/10,000 of the exact value of the integral.
I don't even know what to do in this one! I need some major help! especially in part b and d! DX
Please and Thank You! :)
