Series with positive terms

In summary, the conversation discusses a series with positive terms and a limit of (An+1)/(An) = L < 1. It also shows that for any r < 1, there is a value N such that (An+1)/(An) < r for all n > N. Additionally, it proves that Ak+N is less than or equal to ANr^k for all k, and that (Ak+N)^(N+k) approaches a limit of r as k approaches infinity.
  • #1
jkh4
51
0
Let (infinity)(sigma)(n=1) = An be a series with positive terms such that lim(n -> infinity) = (An+1)/(An) = L < 1

a) Let L < r < 1. Show that there is an N > 0 such that for all n > N, we have (An+1)/(An) < r

b) Show that Ak+N < or = ANr^k for k = 1, 2...

c) Show that lim (k -> infinity) (Ak+N)^(N+k) < or = r

Thanks!

For An+1, it's A with sub n+1
An, it's A sub n
Ak+N is A such (k+N)
ANr^k is A sub N times r^k

Thanks!
 
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  • #2
jkh4 said:
Let (infinity)(sigma)(n=1) = An be a series with positive terms such that lim(n -> infinity) = (An+1)/(An) = L < 1

a) Let L < r < 1. Show that there is an N > 0 such that for all n > N, we have (An+1)/(An) < r
Use the DEFINITION of "limit of a sequence".

b) Show that Ak+N < or = ANr^k for k = 1, 2...
Proof by induction on k.

c) Show that lim (k -> infinity) (Ak+N)^(N+k) < or = r
After b, this should be obvious. What is the limit of rN+k?
 

1. What is a "series with positive terms"?

A series with positive terms is a mathematical sequence in which each term is greater than or equal to zero. It is also known as an "increasing series."

2. How do you determine if a series has positive terms?

To determine if a series has positive terms, you can inspect the terms of the series and see if they are all greater than or equal to zero. Another way is to use the comparison test, which compares the series to a known series with positive terms.

3. What are some examples of series with positive terms?

Some examples of series with positive terms include the geometric series, harmonic series, and p-series. For example, the geometric series 1 + 1/2 + 1/4 + 1/8 + ... has positive terms because each term is greater than or equal to zero.

4. What is the importance of studying series with positive terms?

Series with positive terms have many applications in mathematics and science, such as in calculating probabilities, approximating values, and analyzing growth and decay. Understanding these series can also help in solving more complex mathematical problems.

5. How do you determine if a series with positive terms converges or diverges?

To determine if a series with positive terms converges or diverges, you can use various convergence tests such as the ratio test, root test, or integral test. If the limit of these tests is greater than 1, the series diverges. If it is less than 1, the series converges. If it is equal to 1, the test is inconclusive and you may need to use other methods.

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