1. Determine where the series ∑ fn converges pointwise and wnere it converges uniformly, if fn(x

To determine where the series (sum f_n(x)) converges pointwise and where it converges uniform

Question

1. Determine where the series \( \sum f_{n} \) converges pointwise and wnere it converges uniformly, if \( f_{n}(x) \) is defined as (a) \( \frac{1}{x^{2}+n^{2}} \) (b) \( \frac{1}{x^{n}+1}, x \neq-1 \) (c) \( \frac{x^{n}}{x^{n}+1}, x \neq-1 \) (d) \( \frac{(-1)^{n}}{n+|x|} \) (e) \( \frac{x}{n\left(1+n x^{2}\right)} \).

          1. Determine where the series \( \sum f_{n} \) converges pointwise and wnere it converges uniformly, if \( f_{n}(x) \) is defined as
(a) \( \frac{1}{x^{2}+n^{2}} \)
(b) \( \frac{1}{x^{n}+1}, x \neq-1 \)
(c) \( \frac{x^{n}}{x^{n}+1}, x \neq-1 \)
(d) \( \frac{(-1)^{n}}{n+|x|} \)
(e) \( \frac{x}{n\left(1+n x^{2}\right)} \).

1. Determine where the series ∑ fn converges pointwise and wnere it converges uniformly, if fn(x) is defined as
(a) (1)/(x^2+n^2)
(b) (1)/(x^n+1), x ≠-1
(c) (x^n)/(x^n+1), x ≠-1
(d) ((-1)^n)/(n+|x|)
(e) (x)/(n(1+n x^2)).

Added by Reem A.

Schaum's Outline of Theory and Problems of Advanced Calculus

Robert C. Wrede, Murray Spiegel 2nd Edition

Chapter 11

Instant Answer

Solved by Expert Adi S

07/31/2024

Step 1

Let's go through each case: ### (a) \( f_n(x) = \frac{1}{x^2 + n^2} \) **Pointwise Convergence:** - For each fixed \(x\), as \(n \to \infty\), \(f_n(x) = \frac{1}{x^2 + n^2} \to 0\). Show more…

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