# Series to represent alternate between 1 and -1

• foo9008
In summary, the author of the problem redefines n as 2n'-1 and uses the term (-1)^(n+1) to represent the alternating positive and negative values of the odd terms in the equation. However, it may be confusing because the term (-1)^(3+1) corresponds to n' = 2, not n' = 3 as expected.
foo9008

## Homework Statement

we know that a2 , a4 ,a6 (even number ) = 0 , but when a1 , a3 , a5 (odd numbers) , the answer of an alternate between positive and negative ... in the second circle , the author represent it with (-1)^(n+1) , i don't think this is correct , this is because when n=3 , an = -2/ 3pi when n=3 , [ (-1) ^(3+1 ) ] = positive 1 , not negative 1 ! can someone explain on this ?

## The Attempt at a Solution

I think that the author redefined $n$. To make it clear, you start off assuming that

$f(x) = a_0 + \sum_n a_n cos(nx)$

Then, for this particular problem, you find that for $n > 0$, then $a_n = 0$ unless $n$ is odd. If $n$ is odd, then that means that $n$ can be written as:

$n = 2n'-1$

where $n' = 1, 2, 3, ...$

So the term $-\frac{2}{\pi} \frac{cos(3x)}{1}$ corresponds to $n=3$, but it corresponds to $n' = 2$. In terms of $n'$, the general term is

$\frac{2}{\pi} \frac{cos((2n'-1)x)}{2n'-1} (-1)^{n'+1}$

## 1. How can series be used to represent alternate between 1 and -1?

Series are mathematical representations of a sequence of numbers. In this case, we can use a series to represent the alternating pattern between 1 and -1 by using the alternating sign property, where every other term in the series has a negative sign.

## 2. What is the formula for a series that represents alternate between 1 and -1?

The formula for this series is (-1)^n, where n represents the term number in the series. This formula will result in a series that alternates between 1 and -1, with each term being the opposite sign of the previous term.

## 3. Can series be used to represent other alternating patterns?

Yes, series can be used to represent any alternating pattern. The alternating sign property can be applied to any sequence of numbers to create an alternating series.

## 4. Do series that represent alternate between 1 and -1 always converge?

Yes, the series that represent alternate between 1 and -1 always converge. This is because the series (-1)^n is a geometric series with a common ratio of -1, which is less than 1. This means that as n approaches infinity, the terms in the series will approach 0, resulting in convergence.

## 5. How can series that represent alternate between 1 and -1 be used in real-world applications?

Series that represent alternate between 1 and -1 can be used in various applications, such as signal processing, electronic circuits, and finance. In signal processing, this series can be used to represent alternating currents or waveforms. In electronic circuits, the series can be used to represent the on and off states of a circuit. In finance, the series can be used to calculate the returns on an investment that fluctuates between positive and negative values.

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