Question 9.T.16: Let R be the radius of convergence of the series ∑anx^n. The...

Elements of Real Analysis

Let R be the radius of convergence of the series $\sum{a_{n}x^{n}}.$ Then the sum $S(x) = \Sigma^{∞}_{n=0} a_{n}x^{n}$ is differentiable on (−R, R), and its derivative is given by the series

$S^{\prime}(x) = \sum\limits_{n=1}^{∞}{na_{n}x^{n−1}},$ (9.20)

whose radius of convergence is also R.

The blue check mark means that this solution has been answered and checked by an expert. This guarantees that the final answer is accurate.
Learn more on how we answer questions.

Related Answered Questions

Question: 9.13

Test the following series for convergence: (i) ∑x^n/n (ii) ∑x^n/n² (iii) ∑(sin 1/n) x^n (iv) ∑anx^n, an = { 2^−n n = 2k 3^−n n = 2k + 1. ...

Verified Answer:

(i) With

a_{n} = 1/n,

we have [late...

Question: 9.10

Discuss the convergence properties of the two series (i) ∑sin nx/n² , (ii) ∑sin nx/n. ...

Verified Answer:

(i) Since |sin nx| ≤ 1 for all

x ∈ \mathbb{...

Question: 9.9

Discuss the uniform convergence of the series ∑fn, where (i) fn(x) = sin (x/n²) , (ii) fn(x) = 1/n²x², x ≠ 0, (iii) fn(x) = sin(3^nx)/2^n. ...

Verified Answer:

(i) Suppose D is a subset of

\mathbb{R}[/la...

Question: 9.T.5

Suppose (fn) is a differentiable sequence of functions on [a, b] which converges at some point x0 ∈ [a, b]. If the sequence (f′n) is uniformly convergent on [a, b], then (fn) is also uniformly convergent on [a, b] to a function f which is differentiable on [a, b], and f′n → f′. ...

Verified Answer:

Let ε > 0. From the convergence of

(f_{n...

Question: 9.14

π = 4 ∑n=0^∞ (−1)^n/2n +1. ...

Verified Answer:

Using the Taylor series for the binomial function,...

Question: 9.T.15

Let R be the radius of convergence of the power series ∑anx^n. If 0 < r < R, then the series converges uniformly on [−r, r]. ...

Verified Answer:

For any x ∈ [−r, r],

\left|a_{n}x^{n}\right...

Question: 9.T.14

(Cauchy-Hadamard) The series ∑anx^n is absolutely convergent if |x| R. ...

Verified Answer:

Applying the root test (Theorem 4.8), we obtain [l...

Question: 9.T.13

(Abel’s Test for Uniform Convergence) Suppose the sequences (un) and (vn) of real valued functions defined on D ⊆ R satisfy the following conditions: (i) ∑un converges uniformly on D. (ii) The sequence (vn) is uniformly bounded and monotonically decreasing on D. Then the series ∑unvn is uniformly ...

Verified Answer:

Suppose

|v_{n}(x)| ≤ M

for all [la...

Question: 9.T.12

(Dirichlet Test for Uniform Convergence) Let (un) and (vn) be two sequences of real valued functions on a set D ⊆ R which satisfy the following conditions: (i) The sequence (Un) of partial sums of (un) is uniformly bounded, in the sense that there is a constant K such that ...

Verified Answer:

Let

S_{n} = \Sigma^{n}_{k=1} u_{k} v_{k}.[/...

Question: 9.T.11

(Weierstrass M-Test) Let (fn) be a sequence of functions defined on D, and suppose that there is a sequence of non-negative numbers (Mn) such that |fn(x)| ≤ Mn f or all x ∈ D, n ∈ N. If the series ∑Mn converges, then both ∑fn and ∑|fn| converge uniformly on D. ...

Verified Answer:

Let ε > 0. Since

\sum{M_{n}}

is ...