Questions from Calculus

Q: For what values of p is ∑k=1 ∞ 1

For what values of p is ∑k=1 ∞ 1/kp convergent?

Q: For what values of p is ∑k=1 ∞ 1

For what values of p is ∑k=1 ∞ 1/pk convergent?

Q: Find the Taylor series at x = 0 of the given function

Find the Taylor series at x = 0 of the given function. Use suitable operations on the Taylor series at x = 0 of 1/(1 – x) and ex. 1/(1 + x3)

Q: Find the Taylor series at x = 0 of the given function

Find the Taylor series at x = 0 of the given function. Use suitable operations on the Taylor series at x = 0 of 1/(1 – x) and ex. ln(1 + x3)

Q: Find the Taylor series at x = 0 of the given function

Find the Taylor series at x = 0 of the given function. Use suitable operations on the Taylor series at x = 0 of 1/(1 – x) and ex. 1/(1 - 3x)2

Q: Find the Taylor series at x = 0 of the given function

Find the Taylor series at x = 0 of the given function. Use suitable operations on the Taylor series at x = 0 of 1/(1 – x) and ex. (ex – 1)/x

Q: (a) Find the Taylor series of cos 2x at x

(a) Find the Taylor series of cos 2x at x = 0, either by direct calculation or by using the known series for cos x. (b) Use the trigonometric identity sin2 x = ½ (1 - cos 2x) to find the Taylor series...

Q: (a) Find the Taylor series of cos 3x at x

(a) Find the Taylor series of cos 3x at x = 0. (b) Use the trigonometric identity cos3 x = ¼ (cos 3x + 3 cos x) to find the fourth Taylor polynomial of cos3 x at x = 0.

Q: Let p4 (x) be the fourth Taylor polynomial of f

Let p4 (x) be the fourth Taylor polynomial of f (x) = ex at x = 0. Show that the error in using p4(.1) as an approximation for e0.1 is at most 2.5 * 10-7.

Q: Use the decomposition (1 + x)/(1 – x)

Use the decomposition (1 + x)/(1 – x) = 1/(1 – x) + x/(1 – x) to find the Taylor series of (1 + x)/(1 – x) at x = 0.