# sequences

1. ### Proving the limits of expectations

Hi, I have the following questions. Can anyone show me how to solve the 2 parts? i think it involves using Strong Laws of Large Numbers, Dominated Convergence Theorem, etc. For part (ii), it think the left-hand side can be expressed in the form of an expectation (similar to part (i)). ..
2. ### Proving a Sequence's Convergence

Hello, can you please check my work? Prove sin(n^2)/n^(1/3) converges to 0 Proof for any e > 0 abs(sin(n^2)/n^(1/3)) < e sin(n^2)/n^(1/3) < e Since the range of sin(n^2) is [-1,1], sin(n^2)/n^(1/3) < or = 1/n^(1/3) so let 1/n^(1/3) < e. So 1/n < e^3 let N < or = n Then, sin(n^2)^3/n < or =...
3. ### Sequences Question Help

Hello all, I'm really stuck on this question and can't find anything that can help me figure it out. Any help and explanation or links to stuff is greatly appreciated thank you!! "A bank account starts with £300. During the ﬁrst week after opening the account, an additional £10 is added to the...
4. ### Arithmetic and Geometric Sequences.

Hi, I just did these problems for my final review. Does anyone mind checking them? Tyvm. 1) Find the 98th term of an arithmetic sequence whose first three terms are 2,6,10 Ans: 390 2)Write the 21st term in arithmetic sequence with a first term of 7, and a common difference of 5. Ans: 107 3)...
5. ### Help me to find the little mistake in this demostration about Cauchy sequences

Help me to find the little mistake in this demonstration about Cauchy sequences My Real Analysis teacher told us that there is a mistake at the end of this demonstration. I am supposed to find it. Can you help me please? It's in Spanish but we all know that the mathematical language is...

7. ### Using the sandwich theorem for sequences

Instead of using bn = 1/(6n-5) and cn = -1/(6n-5) like I did here, could I have used bn = 0 and cn ​= 1/n? Those would get me 0 when evaluated using the limit as n approaches infinity.
8. ### Help: Real analysis and Bounded/ divergent/ monotonic sequences.

Hi! I'm Gabriel. I'm from Honduras. I need to give an example of a sequence with the following characteristics: 1 - Monotonic and divergent. 2 - Bounded and divergent. 3 - It must have two subsequences converging to different numbers. I'm taking Real Analysis in the University. Thank you...
9. ### Determine whether the sequence converges or diverges

I have the sequence: I found that it converges to 0, and I checked in the back of the book and that was correct, but I want to make sure my reasoning is correct. I took the highest powers to be e^n/e^2n which is just equal to e^-n, so the limit as n goes to infinity would just be 0. I'm not...
10. ### Geometric Sequences

How many terms of the geometric sequence 1.2, 2.4, 4.8, 9.6, 19.2...are required for the sum to be greater than 10,000? Do you use the equation sn=a(1-r^n)/1-r? And how is logarithms included in this question?
11. ### Help with Sequences

So the question asks: find if the sequence diverges or converges, and to find the limit if it converges. a_n = \left (\frac{x^n}{2n+1} \right )^{\frac{1}{n}} (assume x>0) ln a_n = \frac{1}{n} \ln \left ( \frac {x^n}{2n+1} \right ) ln a_n = \frac { \ln \left ( \frac {x^n}{2n+1} \right )}{n}...
12. ### Help needed with factoring in a sequences problem

Hi, The following link is for a sequences problem, I've been doing. Link: Mathopolis Question Database I got most of the way to finding the solution shown by the multiple choice answers but got stuck at some factoring that was required. Here's what I got to on my own without looking at the...
13. ### Telescoping Sums and Convergence

So I'm pulling an all nighter when I came across a problem that I've been stuck on for a while; I have to determine if a series is convergent or divergent and find its sum I found it's convergent and wrote a telescoping series as shown here; SUM(n=0,infinity) (1/(5n^2 +2n -3)) Telescoping...
14. ### Sequences and Series

Hi, I uploaded questions (c and d), and my answers. I am not sure if I did them correctly, especially question d. Can anyone please help me with these questions and whether I got them right. Thank you, ALexeia
15. ### Sequences and Summations: Deduce a closed formula

Please refer to the image, which contains my question in red ink.
16. ### Sequences, show that {tn} is a decreasing sequence.

Kind of lost as to what I am supposed to do here. I am given that tn=1 + 1/2 + 1/3 +...+1/n - ln(n) The question is: Interpret tn - tn+1= [ln(n+1)-ln(n)] - 1/(n+1) as a difference of areas to show that tn - tn+1 > 0. Therefore, {tn} is a decreasing sequence. First of all isn't there an 1/n...
17. ### Detremine whether these sequences converge or diverge

I need to determine whether these sequences converge or diverge and if they do converge, to what limit. a) an=(3n^2 + 4^n)/3^n -5 , n = 1, 2,... b) an=( 3(n^2 + 1)(n^2 − 2) )/(2n + 1)^3(n + 3) I think that a) diverges, but I'm not sure how to show this. I know sequences are divergent if...
18. ### Generalized Holder Inequality

Let $a_i \in \mathbb R^n$ with $a_i = (a_{i}^j)_{j = 1 ... n} = (a_{i}^1, ... ,a_{i}^n)$ for $i = 1, ... , k$ and let $p_1,...,p_k \in \mathbb R_{>1}$ with $\frac1{p_1}+ ... + \frac1{p_k} = 1$ Then show the following inequality by assuming that there are for every $i = 1, ... ,k$ one \$N \in...
19. ### Sandwich theorem for sequences - question help

Hi, I'm trying to find a limit for the sequence an=(3n+1)1/n How would I go about defining the upper and lower limits for a sequence in this form? Any help would really be appreciated! (Happy)
20. ### Methods to Solve a number sequences

There are many number sequences that we get to solve, specially through the internet. One of the number series are : 1 2 6 24 120 720 After spending some time I figured it out that the series is 1 x 2 = 2 2 x 3 = 6 6 x 4 = 24 24 x 5 = 120 120 x 6 = 720 Another series is 3 9 8 24 23...