Question

Suppose that 0 14 years ago

Answer 1

The series is convergent because 0<a<Xn<b but I can't find the limit.

Answer 2

Nah, your argument is invalid. What theorem have you used to confirm the result?

Answer 3

@blockcolder how do you know that sequence is converge???can you prove it..

Answer 4

We have
\[ x_{n+1}-x_n=\frac{a^{n+2}+b^{n+2}}{a^{n+1}+b^{n+1}}-\frac{a^{n+1}+b^{n+1}}{a^n+b^n} \\
=\frac{a^nb^n(a-b)^2}{(a^{n+1}+b^{n+1})(a^n+b^n)} \ge 0\]
Therefore,
\[ x_n \hspace{3mm} \textrm{is an increasing sequence} \]
Moreover,
\[ x_n = \frac{a^{n+1}+b^{n+1}}{a^n+b^n} <a+b \hspace{3mm} \forall n \in \mathbb{N} \]
By Monotone Convergence Theorem, we have
\[ \langle x_n \rangle \hspace{3mm} \textrm{ is a convergent sequence} \]

Answer 5

To find the limit , we do the following step
\[ \lim_{n \to \infty}x_n=\lim_{n \to \infty}\frac{a^{n+1}+b^{n+1}}{a^n+b^n}=\lim_{n \to \infty}\frac{a\left(\frac{a}{b}\right)^n+b}{\left(\frac{a}{b}\right)^n+1}=b \]

Answer 6

by monotone convergence theorem,if {Xn} is bounded above and increasing, then {Xn} is converge....but how do you know Xn bounded???is it because of 0<a<b????

Answer 7

As I have proved that for all n natural numbers, we have
\[ x_n <a+b \]
Therefore,
\[ x_n \hspace{3mm} \textrm{is bounded above} \]

Answer 8

oooo....i understand.....thank you for helping me.....but if Xn=(a^n+b^n)^1/n can we use the same theorem or using p-series?????

Answer 9

Interesting enough, for the second question, we can basically show that the limit is equal to 0 directly using Sandwich Theorem ( Squeeze Theorem)

Answer 10

okk...i will try to solve it....thanks again....

Answer 11

No worry. Good luck.

Answer 12

Oh wait, sorry for the misleading answer , the answer is again b. Not 0 as noted above. Truly apologize for the fatal mistake and yes, we use the same technique as I did above.

Answer 13

We have
\[ \lim_{n \to \infty}x_n =\lim_{n \to \infty}(a^n+b^n)^{\frac{1}{n}}=\lim_{n \to \infty}b\left(\left[\frac{a}{b}\right]^n+1\right)^n=b \]

Answer 14

its okay....thanks...

Answer 15

thx..