Limits/ Series

[FrozenDragon427]

Can someone help me with these 3 questions?

 

[Deleted member 4993]

Can someone help me with these 3 questions?

Please show us what you have tried and exactly where you are stuck.

Please follow the rules of posting in this forum, as enunciated at:

https://www.freemathhelp.com/forum/threads/read-before-posting.109846/#post-486520​

Please share your work/thoughts about these problems.

 

[Steven G]

Come on you can take a number between 0 and 1, like 1/2, keep multiplying by itself and see where this is going? You have to try!

1/2: 1/2, 1/4, 1/8, 1/16, 1/32, 1/64, 1/128, ...
1/3: 1/3, 1/9, 1/27, 1/81, 1/243, 1/729, 1/2187, ...,
1/7: 1/7, 1/49, 1/343, 1/2407, 1/16849, ...

 

[FrozenDragon427]

Okay so the thing is I will do these topics (Series, Limits, Logarithms) in school, but I have a competition coming up that requires knowledge of all the maths, so I'm trying to cover up as much as I can in a week.

 

[FrozenDragon427]

Can someone help me?

 

[pka]

There is no reason to complicate matters.
If \(0<r<1\) then \(0<r^{n+1}<r^n<1\) i.e. If \(0<r<1\) then \(r^n\) forms a decreasing sequence and has limit \(0\).
\(\begin{gathered}
{S_N} = \sum\limits_{n = 0}^N {a{r^n}} = a + ar + a{r^2} + \cdots + a{r^N} \hfill \\
r{S_N} = ar + a{r^2} + \cdots + a{r^N} + a{r^{N + 1}} \hfill \\
{S_N} - r{S_N} = a + ar + a{r^2} + \cdots + a{r^{N + 1}} \hfill \\
{S_N} = \frac{{a - {r^{N + 1}}}}{{1 - r}}\quad \mathop {\lim }\limits_{N \to \infty } {S_N} = \frac{a}{{1 - r}} \hfill \\ \end{gathered} \)
So in #40 what is the answer?

 

[FrozenDragon427]

There is no reason to complicate matters.
If \(0<r<1\) then \(0<r^{n+1}<r^n<1\) i.e. If \(0<r<1\) then \(r^n\) forms a decreasing sequence and has limit \(0\).
\(\begin{gathered}
{S_N} = \sum\limits_{n = 0}^N {a{r^n}} = a + ar + a{r^2} + \cdots + a{r^N} \hfill \\
r{S_N} = ar + a{r^2} + \cdots + a{r^N} + a{r^{N + 1}} \hfill \\
{S_N} - r{S_N} = a + ar + a{r^2} + \cdots + a{r^{N + 1}} \hfill \\
{S_N} = \frac{{a - {r^{N + 1}}}}{{1 - r}}\quad \mathop {\lim }\limits_{N \to \infty } {S_N} = \frac{a}{{1 - r}} \hfill \\ \end{gathered} \)
So in #40 what is the answer?

C

 

[Steven G]

Can someone help me?

Sure. Can you please look at my post #3 and conclude what lim(n=>infinity) (x^n) equals if 0<x<1? Just think what happens to a number between 0 and 1 when you keep multiplying it by itself. Just take a calculator, type in .5 and keep hitting the times button. Try it with .714, .001, etc. What do you eventually get?

 

[Steven G]

41) Let y = e_ln(n).
Now take ln of both sides and simplify. What does y equal?