Prove that series converges uniformly

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Thus, the $n$th term is dominated by $\dfrac{1}{2^{n-1}}$, which is a convergent geometric series with ratio $1/2$. Therefore, the given series converges uniformly on $S=[2, \infty)$ by the Weierstrass M-test.
  • #1
fabiancillo
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I was studying uniform convergence. I have doubts

a) Prove that series $\displaystyle\sum_{n=1}^\infty{\displaystyle\frac{ln(1+nx)}{nx^n}}$
converges uniformly on the set $ S = [2, \infty) $.

b Prove that series $\displaystyle\sum_{n=1}^\infty{(-1)^{n+1} \displaystyle\frac{e^{-nt}}{\sqrt[ ]{n+t^2}}}$
converges uniformly on the set $ S = [0, \infty) $.

My attempt:

a) The fuctions $f_n(x)=\displaystyle\frac{ln(1+nx)}{nx^n}$ are decreasing (I don't how prove). Therefore $|f_n(x)|\leq f_n(2)$

b) I do not know how to start

Thanks
 
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  • #2
Apply the Weierstrass M-test in series (a). Use the inequality $\ln(1+t) \le t$ for all $t \ge 0$ to show that the $n$th term is dominated by $\dfrac1{2^{n-1}}$.

For (b), note that the series is alternating. So it suffices to show that $\dfrac{e^{-nt}}{\sqrt{n+t^2}}$ is decreasing (by showing its derivative is negative) and converges uniformly to $0$.
 
  • #3
Ok but I don't understand how to test this:

show that the $nth$ term is dominated by
$\dfrac1{2^{n-1}}$
 
  • #4
Let $x\ge 2$. Since $\ln(1+ nx)\le nx$, then $$\frac{\ln(1+nx)}{nx^n}\le\frac{nx}{nx^n} = \frac{1}{x^{n-1}} \le \frac{1}{2^{n-1}}$$
 

FAQ: Prove that series converges uniformly

What is the definition of uniform convergence of a series?

The uniform convergence of a series is a type of convergence where the convergence is independent of the choice of the point in the domain.

How is uniform convergence different from pointwise convergence?

Uniform convergence guarantees that the convergence occurs at the same rate for all points in the domain, while pointwise convergence only guarantees convergence at each individual point in the domain.

What is the Cauchy criterion for uniform convergence?

The Cauchy criterion states that a series converges uniformly if for any given positive number ε, there exists a positive integer N such that for all n > N and all x in the domain, the absolute value of the difference between the nth partial sum and the (n+1)th partial sum is less than ε.

How can I prove that a series converges uniformly?

To prove that a series converges uniformly, you can use the Cauchy criterion or the Weierstrass M-test. You can also show that the series satisfies the definition of uniform convergence by showing that the difference between the partial sums and the limit function approaches 0 as n approaches infinity.

What are some common examples of series that converge uniformly?

Some common examples of series that converge uniformly include geometric series, power series, and Taylor series. Other examples include series of polynomials and rational functions.

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