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.

Advanced Calculus: series of functions. ...series n=1 to infinity (-1)^(n-1)* x^n = x- x^2+ x^3 - x^4+....D ... (Do not rely on a general fact about power series.). ...

Complex Function Calculus. ... domain while the function is also analytic everywhere but it equals zero at hence the function has poles at The Taylor series of the ...

... Expanding ln(1+x) using Taylor series and find the range of x. The following series is developed in calculus: ln (l + x) = x - (x^2/2) + (x^3/3) - (x^4/4 ...

Need Assistance Understanding Taylor and Maclaurin Series. ... I am in a fast-paced Calculus course where I must ... to grasp the concept of the Taylor and Maclaurin ...

...Series. How do I find the Taylor series for this function: ... 1)!]+. Note: We use the fundamental theorem of Calculus to get the integral above! ...

... 3. Truncation error is the error due to truncating the Taylor series (above). ... provides step-by-step explanations for how to solve the given calculus problems. ...

... A calculus instructor uses computer aided instruction and allows ... the difference in successive values of a time series. ... p^2 (^ stands for raised to the power of ...

... by the use of tensor calculus, to the ... Thus information processing and decision-making power should be ... founded the domain through a series of interdisciplinary ...