The series
eˣ = 1 + x + x²/2! + x³/3! + x⁴/4! + x⁵/5! + ⋯
converges to eˣ for all x.
a. Find a series for (d/dx)eˣ. Do you get the series for eˣ? Explain your answer.

∑ (from n=1 to ∞) (1 / √(n + 1)) diverges

b. What should n be in order that the partial sum sₙ = ∑ (from i=1 to n) (1 / √(i + 1)) satisfies sₙ > 1000?

(Continuation of Exercise 61.) Use the result in Exercise 61 to determine which of the following series converge and which diverge. Support your answer in each case.

a. ∑ (from n=2 to ∞) [1 / (n ln n)]

The series

sec x = 1 + x²/2 + 5x⁴/24 + 61x⁶/720 + 277x⁸/8064 + ⋯

converges to sec x for −π/2 < x < π/2.

a. Find the first five terms of a power series for the function ln|sec x + tan x|. For what values of x should the series converge?

Quadratic Approximations The Taylor polynomial of order 2 generated by a twice-differentiable function f(x) at x = a is called the quadratic approximation of f at x = a. In Exercises 41–46, find the (a) linearization (Taylor polynomial of order 1)

f(x) = ln(cos x)

Intervals of Convergence

In Exercises 1–36, (a) find the series’ radius and interval of convergence.

∑ (from n = 1 to ∞) [ (√(n + 1) − √n)(x − 3)ⁿ ]

Intervals of Convergence

In Exercises 1–36, (a) find the series’ radius and interval of convergence.

∑ (from n = 0 to ∞) [ (−2)ⁿ (n + 1) (x − 1)ⁿ ]