Series Identities: Showing Convergence to cL & X+Y

In summary, the first part of the problem asks to show that if a series is convergent to a limit L, then multiplying each term by a constant c results in a new series that is also convergent to cL. The second part asks to show that the sum of two convergent series is also convergent, with the limit being the sum of the individual limits. The definition of convergence for a series is used in both parts, and the first part also requires understanding of Delta-Epsilon proofs of limits of sequences.
  • #1
dtl42
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Homework Statement


a) If c is a number and [tex]\sum a_{n}[/tex] from n=1 to infinity is convergent to L, show that [tex]\sum ca_{n}[/tex] from n=1 to infinity is convergent to cL, using the precise definition of a sequence.

b)If [tex]\sum a_{n}[/tex] from n=1 to infinity and [tex]\sum b_{n}[/tex] from n=1 to infinity are convergent to X and Y respectively, show that [tex]\sum b_{n}+a_{n}[/tex] from n=1 to infinity is convergent to X+Y.

Homework Equations


I personally thought these were identities, and have no idea how to approach them.


The Attempt at a Solution


a) Maybe [tex]\sum a_{n}[/tex] from n=1 to infinity = [tex] Lim (S_{n}) [/tex] as n goes to infinity, has something to do with it


I cross posted this in Calculus & Beyond and Pre-Calculus because I wasn't sure where it belongs
 
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  • #2
I bet the problem does NOT say "using the precise definition of a sequence". I'll bet it says "using the precise definition of convergence of a series". Write down the definition of convergence of a series as it applies to these two series. You are told that one converges and should be able to use that to show that the other converges.

"I personally thought these were identities, and have no idea how to approach them." I have no idea what you mean by that! Were you under the impression that one doesn't prove identities?
 
  • #3
The problem actually does say "Use the precise definition of limits for sequences in Sec. 10.2", that section covers Delta-Epsilon proofs of limits of sequences.
 

1. What is a series identity?

A series identity is a mathematical expression that represents the sum of an infinite sequence of numbers. It is often used to calculate the limit of a sequence as it approaches a fixed value.

2. How is convergence to cL and X+Y determined in series identities?

In series identities, convergence to cL and X+Y is determined by taking the limit of the sum of the terms as the number of terms approaches infinity. If the limit is equal to a fixed value, cL, then the series converges. If the limit is equal to X+Y, then the series is conditionally convergent.

3. What is the difference between absolute and conditional convergence in series identities?

Absolute convergence in series identities means that the series converges regardless of the order in which the terms are added. Conditional convergence means that the series only converges if the terms are added in a specific order.

4. Can series identities be used to prove the convergence of all infinite series?

No, series identities can only be used to prove the convergence of certain types of infinite series. They are most commonly used for power series, which are infinite series of the form a0 + a1x + a2x2 + a3x3 + ..., where x is a variable and an is a constant coefficient.

5. How are series identities applied in real-world situations?

Series identities are used in various fields of science and engineering to model and analyze real-world phenomena. They are commonly used in physics and engineering to calculate the behavior of systems that involve continuous change, such as electrical circuits and fluid flow. They are also used in statistics and probability to calculate the probability of events occurring over an infinite number of trials.

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