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Taylor Series Calculus

The starting point for this example is the Taylor series for sine:
sin (x) = x - (1/3!)x^3 + (1/5!)x^5 ?(1/7!)x^7 + ...... + (-1)^n * 1/(2n + 1)!*x^(2n+1) + .......
a) Let f(x) = { sin(x) / x if x doesn't equal 0
{1 if x doesn't equal 0

Show that f is infinitely differentiable on R (including x = 0) and determine its Taylor series in powers of x.
b) Define the function Si by the rule,
Si (x) = the integral from 0 to x
sin(t) / t dt , x E R
Determine the Taylor series for Si in powers of x, and the open interval of convergence of the resulting series.

Solution Preview

Please find the attachment for the solutions.

The starting point for this example is the Taylor series for sine:
sin (x) = x - (1/3!)x^3 + (1/5!)x^5 -(1/7!)x^7 + ...... + (-1)^n * 1/(2n + 1)!*x^(2n+1) + .......
a) Let f(x) = { sin(x) / x if x doesn't equal 0
{1 if x doesn't equal 0

Show that f is infinitely differentiable on R (including x = 0) and determine its Taylor series in powers of x.
b) Define the function Si by the rule,
Si (x) = the integral from 0 to x
sin(t) / t ...

Solution Summary

The solution provides an example of the Taylor series caluclus.

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