Using Weiestrass M-test, show that the following series converges uniformly.
Consider the series (\sum_{n=1}^{\infty} n^3 x^n) for (x \in \left[-\frac{1}{3}, \frac{1}{3}\right]). To apply the Weierstrass M-test, we need a sequence of constants (M_n) such that (|n^3 ___ ___ _________ _______ _____ _______ ___.
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a) Test the following series for convergence,
(i)
(ii)
b) Show that is conditionally convergent.
Use the Fundamental Theorem of Integral Calculus to evaluate the integral
Show that , the lagrange's form of remainder in the Maclaurin series expansion of e4x, tends to zero as
. Hence obtain the Maclaurin's infinite expansion for e4x.
Find the following limit
a) If the partition P2 is a refinement of the partition P1 of [a,b], then and
. Verify this result for the function
defined over the interval
and the partitions
and
.
b) Evaluate: .
Prove that a strictly decreasing function is always one-one.
Use Cauchy's mean value theorem to prove that: