Finding the sum of an infinite series

• szklany
In summary, to find the sum of the infinite series, you can use the formula for geometric series and consider the simplification of (i/2)^2. You can also use Euler's identity and the property of cos(nπ) to simplify the series. Ultimately, the series can be represented as a purely real geometric series by using the simplified form of (i/2)^2.
szklany
Find the sum of the infinite series

$$\sum _{n=1}^{\infty } \left( i /2\right) ^{2\,n}$$

I just can't seem to get started on this problem, so I was hoping somebody could give me a hint, as to what methods i should read up on.

A good place to start would be to consider what (i/2)^2 is, that simplifies the sum considerably. You also need the formula for geometric series, that you can find for example from wikipedia.

Actually, break i into its polar form which is $$i^2^n = e^i^n^(^p^i^)$$ . Now expand using Euler's identity and you are left only with $$cos\ n(pi) = (-1)^n$$ From there it is a geometric series.

Actually, there is no need to worry about "i". $(i/2)^{2n}= (i^2/4)^n= (-1/4)^n$ so this is a purely real geometric series.

1. What is an infinite series?

An infinite series is a sum of an infinite number of terms. Each term in the series is added to the previous one to create a sum.

2. How do you find the sum of an infinite series?

The sum of an infinite series can be found using various methods such as the geometric series formula, telescoping series, or the ratio test.

3. What is the difference between a finite and infinite series?

A finite series has a limited number of terms, while an infinite series has an infinite number of terms. This means that the sum of a finite series will eventually reach a final value, while the sum of an infinite series will continue to increase without ever reaching a final value.

4. Can an infinite series have a finite sum?

Yes, an infinite series can have a finite sum if the terms in the series decrease in value as the series progresses. This is known as a convergent series.

5. What is the importance of finding the sum of an infinite series?

Finding the sum of an infinite series is important in various fields such as mathematics, physics, and engineering. It allows us to calculate and predict the behavior of systems that involve infinite sums, such as electric circuits and population growth.

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