Converging Series: Solve Sum of (3^n + 4^n) / (3^n + 5^n)

In summary, a converging series is a sequence of numbers that approaches a specific limit as more terms are added. To determine if a series is converging or diverging, tests such as the Ratio Test, the Root Test, or the Comparison Test can be used. The formula for the sum of an infinite series is S = a/(1-r), but it only works for geometric series with a common ratio between -1 and 1. To solve a sum like (3^n + 4^n) / (3^n + 5^n), the Ratio Test can be used to determine convergence. An example of a converging series is the geometric series 1/2 + 1/4 + 1/8 + 1
  • #1
kmeado07
40
0

Homework Statement



Show that the following series converges:

Homework Equations



Sum of (from n=1 to infinity) of [3^n + 4^n] / [3^n + 5^n]

The Attempt at a Solution



Some help on this question would be much appreciated as i really don't know how to start it. Thanks
 
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  • #2
What i would do is break the series into two. then by observing that:


[tex]3^n+5^n>5^n=>\frac{1}{3^n+5^n}<\frac{1}{5^n}=>\frac{3^n}{3^n+5^n}<\left(\frac{3}{5}\right)^n[/tex]


Also:

[tex]3^n+5^n>5^n=>... \frac{4^n}{3^n+5^n}<\left(\frac{4}{5}\right)^n[/tex]
 

Related to Converging Series: Solve Sum of (3^n + 4^n) / (3^n + 5^n)

1. What is a converging series?

A converging series is a sequence of numbers that approaches a specific limit as more terms are added. In other words, the sum of the terms in the series will get closer and closer to a specific value as the number of terms increases.

2. How do you determine if a series is converging or diverging?

To determine if a series is converging or diverging, you can use a variety of tests such as the Ratio Test, the Root Test, or the Comparison Test. These tests compare the given series to a known convergent or divergent series to determine its behavior.

3. What is the formula for the sum of an infinite series?

The formula for the sum of an infinite series is given by S = a/(1-r), where S is the sum, a is the first term, and r is the common ratio between consecutive terms. However, this formula only works for geometric series with a common ratio between -1 and 1.

4. How do you solve the sum of (3^n + 4^n) / (3^n + 5^n)?

To solve this sum, we can use the Ratio Test. We can rewrite the series as (4/3)^n + (5/3)^n and then compare it to the convergent geometric series (5/3)^n. Since the ratio between consecutive terms in our series is always less than 1, the series converges.

5. Can you provide an example of a converging series?

One example of a converging series is the geometric series 1/2 + 1/4 + 1/8 + 1/16 + ... This series approaches a limit of 1 as the number of terms increases.

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