Find the Sum of the Series SUM((2^n + 1)/3^n)

TL;DR

Learn how to find the sum of an infinite geometric series by breaking it down and applying the common ratio formula.

Transcript

find the sum of the series so the way we're going to do this is we are going to break it up so note that if you have 2 to the n plus 1 over 3 to the n we can write this as 2 to the n over 3 to the n plus 1 over 3 to the n then we can take a step further and use properties of exponents to write this as 2 over 3 and the whole thing is to the nth powe... Read More

Key Insights

• 🥳 The sum of an infinite geometric series can be found by breaking it into smaller series and applying the common ratio formula.
• 🥳 The convergence of a series is determined by the absolute value of the common ratio being less than one.
• 🥳 The common ratio formula involves plugging in the initial term and dividing by 1 minus the common ratio.
• ✊ The properties of exponents, such as the power of a quotient property, can be used to simplify the series.

Questions & Answers

Q: How do you find the sum of an infinite geometric series?

To find the sum, first identify the common ratio by dividing a term by its preceding term. Then, use the common ratio formula by plugging in the initial term and dividing by 1 minus the common ratio.

Q: What property of exponents is used to simplify the series?

The property of exponents used is the power of a quotient property. It allows us to write 2 to the n over 3 to the n as 2 over 3 to the n.

Q: Why are both series convergent?

Both series are convergent because the absolute value of the common ratio is less than one. In one series, the common ratio is 2/3, and in the other series, it is 1/3.

Q: What happens if there is a 0 or a number other than 1 as the starting term?

In the common ratio formula, you simply substitute the starting term for "n" and divide the result by 1 minus the common ratio.

Summary & Key Takeaways

• The content teaches how to find the sum of an infinite geometric series by breaking it into two separate series.

• Both series are convergent because the absolute values of their common ratios are less than one.

• To find the sum, use the common ratio formula by plugging in the initial term and dividing by 1 minus the common ratio.