Cauch ' s form a ≤ ξ ≤ x This result holds if f(x) has continuous derivatives of order n at last. If lim R = 0 , the infinite series obtained is called n n →∞ Taylor series for f(x) about x = a. If a = 0 …

In this series the sizes of the terms decrease, that is, |an| forms a decreasing sequence, but this is not required in an alternating series. As with positive term series, however, when the terms do …

Here's a list of all of the convergence tests for series that you know so far: Divergence test (a.k.a. n-th term test) Geometric series test Telescoping series Integral test

Convergence and Divergence Tests for Series. for Series Test When to U. re f(n) = an. Comparison Test ∞ . n=0 n=0 if 0 ≤ an ≤ bn ∞ �. n=0 ∞ ∞ X X a. diverges =⇒ bn d. n ∞ ∞ X X …

Power series are good locally, but diverge globally. Rational approximations can achieve a global, uniform convergence.

For what values of x does the power (a.k.a. Taylor) series. P1(x) = X f(n)(x0) (x x0)n n!

Answer: The terms √ 1 are decreasing and go to zero (you should check this), so the Alternating n2+1 Series Test says that the series converges. To see that the series does not converge …