Sequence and series Definition and 22 Threads

Below is a screenshot of an insurance policy: I was told by the insurance company that the return becomes lower and lower with the age till death increasing. And this reduction in return is by an annuity ratio of 6%. I wanted to find out the formula so that I can understand the effect of...

We're given the series ##\sum_{n=1}^{\infty} [ \sqrt{n+1} - \sqrt{n} ]##. ##s_n = \sqrt{n+1} - 1## ##s_n## is, of course, an increasing sequence, and unbounded, given any ##M \gt 0##, we have ##N = M^2 +2M## such that ##n \gt N \implies s_n \gt M##. Thus, the series must be divergent. But...

Homework Statement:: Tell me if a sequence or series diverges or converges Relevant Equations:: Geometric series, Telescoping series, Sequences. If I have a sequence equation can I tell if it converges or diverges by taking its limit or plugging in numbers to see what it goes too? Also if I...

It is clear that the terms of the sequence tend to zero when n tends to infinity (for some α) but I cannot find a method that allows me to understand for which of them the sum converges. Neither the root criterion nor that of the relationship seem to work. I tried to replace ##\sqrt[n]{n}## with...

Problem: If sequence ## (a_n) ## has ##10-10## as partial limits and in addition ##\forall n \in \mathbb{N}.|a_{n+1} − a_{n} |≤ \frac{1}{n} ##, then 0 is a partial limit of ## (a_n) ##. Proof : Suppose that ## 0 ## isn't a partial limit of ## (a_n) ##. Then there exists ## \epsilon_0 > 0 ## and...

I think the initial assumptions would allow me to prove this without induction. Suppose ##(x_n)## is a real sequence that is bounded above. Define $$ y_n = \sup\{x_j | j \geq n\}.$$ Let ##n \in \mathbb{N}##. Then for all ##j \in \mathbb{N}## such that ##j \geq n + 1 > n## $$ x_{j} \leq y_n.$$...

Hi, this is my try:Thanks.

I tried by ##S=1+(1/1!)(1/4)+(1.3/2!)(1/4)^2+...## ##S/4=1/4+(1/1!)(1/4)^2+(1.3/2!)(1/4)^3..## And then subtracting the two equations but i arrived at nothing What shall i do further?

The pth term of an airthmetic progression is 1/q and qth term is 1/p. What is the sum of pqth term?

First, I would like to clear up notation and the definition for sequences. What exactly is a sequence? I read somewhere that it is defined as a function ##f: \mathbb{N} \to \mathbb{R}##. But if this is the case, why do we only define functions based on the range of the function, e.g., ##\left \{...

Homework Statement find the sum of the series to 'n' terms (1^3 / 1 ) + (1^3 + 2^3 / 1+2 ) + (1^3 + 2^3 + 3^3 / 1+2+3 ) +... Homework Equations sigma n^2 = n(n+1)(2n+1) / 6 sigma n = n(n+1) / 2 The Attempt at a Solution numerator = 1^3 + 2^3 + ... n^3 denominator = 1+2+3+...n...

Homework Statement Two consecutive numbers from 1,2,3...n are removed A.M of remaining numbers is 105/4. Find n and those numbers removed .Homework Equations Answer n = 50 those numbers are 7 and 8 The Attempt at a Solution I solved this question like a few weeks ago but now it escaped my...

Hi! While studying sequence and series, I've gotten some misunderstandings in the definitions of sequence and series. What I know about the definitions of sequence and series is as follows below ; a sequence of field of real numbers is defined as a function mapping of the set of all positive...

Homework Statement Find all positive integers n such that both n + 2008 divides n^2 + 2008 and n + 2009 divides n^2 + 2009 Homework Equations - The Attempt at a Solution I have no idea where to start... I'm not even sure it's a sequence and series question. If it is then I have...

After studying Generating Functions some, I wondered this question: 1). Take an arbitrary sequence of numbers for example: {p1,p2,p3,p4...pN} and get/find a Generating Function for that sequence. 2). Now 'remove' an arbitrary sequence members from the above set :{p1,p2,p3,p4...p100}...

is there a way to represent the sequence 1*3*5*7*9...(2n-1) for n>=1 with a factorial if not is thre a way to input that sequence ino a calculator with somthing

Got a couple questions please help! 1. lim n>inf (x^4)/(e^(-2x)) How do you do this one? i know you use L hospital rule but you can never get rid of the denominator. or is there a trick i am missing? 2. Determine whether the sequences are increasing, decreasing, or not monotonic...

Homework Statement Let xn = 1/ln(n+1) for n in N. a) Use the definition of limit to show that lim(xn) = 0. b) Find a specific value of K(ε) as required in the definition of limit for each of i)ε=1/2, and ii)ε=1/10. The Attempt at a Solution a) If ε > 0 is given, 1/ln(n+1) < ε <=>...

Homework Statement 0<y0<x0 x1=(x0+y0)/2 y1=\sqrt{x0y0} in general Xn+1=(xn+yn)/2 Yn+1=\sqrt{XnYn} Homework Equations none The Attempt at a Solution I have no idea I tried to solve for Xn and substituting that into another equation... however, I don't know how to simplify...

There was this awesome help site for this topic. The name of it was something like "Dave's math help". I thought I saved the link but guess I didn't. Anyone know where I could possibly find this site or know of others?

Tod injured his wrist while he was skateboarding. To relive the pain his doctor prescribed him some medication. He takes 500mg of medicine every 6h. Only 25 % of the medication remains in his body by the time he is ready to take another pill. Write and equation to model this. so 25% is...

Right here is my sequence 2, 5, 8, 11, 14, ... I have been asked to prove that the cube of any number in the sequence is in the sequence. my answer: General term: a_n=3n+2 We need to cube a_n and see if it matches a number in the series i.e. (a_n)^3 = 3q+2 for some integer q...