Understanding the Divergence of the Infinite Series 1 - (-1)^n

The series S sum_{n0}^{infty} 1 - (-1)^n is an interesting example of an infinite series that does not converge to a finite value. This article will explore the behavior of this series, its divergence, and the various methods to understand why it diverges to infinity.

Behavior of the Series Term

To analyze the infinite series S sum_{n0}^{infty} 1 - (-1)^n , we first break it down based on the behavior of the term 1 - (-1)^n for even and odd values of n.

Even and Odd Values of n

When n is even, i.e., n 0, 2, 4, ldots :

When n 2k (where k is an integer), -1^{2k} 1. Therefore, 1 - (-1)^n 1 - 1 0, but the series has a term 1 before this, so the term effectively becomes 1 - (-1)^n 2.

When n is odd, i.e., n 1, 3, 5, ldots :

When n 2k 1 (where k is an integer), -1^{2k 1} -1. Therefore, 1 - (-1)^n 1 - (-1) 2.

Rewriting the Series

With this understanding, the series can be rewritten as:

S 2 0 2 0 2 0 ldots

Here, the series alternates between the terms 2 and 0. Specifically, for every even n, the series contributes 2, and for every odd n, the series does not contribute to the sum, but the series alternates between 2 and 0 effectively contributing 2 at each even step.

Divergence of the Series

Given the alternating pattern of 2 and 0, the sum of the series can be derived as:

S 2 0 2 0 2 0 ldots

Thus, the series consists of infinitely many terms of 2 when n is even. The sum of the series grows indefinitely, and it does not converge to a finite value. Instead, it diverges to infinity:

S infty

Formal Proof of Divergence

To formally prove the divergence, we use the convergence test. The necessary condition for a series to converge is that the limit of its terms must be zero.

[lim_{n to infty} (1 - (-1)^n)]

This limit does not exist. Therefore, the series does not converge, and hence it must diverge to infinity.

Calculation of Partial Sums

We can also use the definition of the sum of a series by calculating the limit of its partial sums. The table below shows the partial sums for various values of n: | n | a_n | S_n | |---------|--------------|------------| | 0 | 2 | 2 | | 1 | 0 | 2 | | 2 | 2 | 4 | | 3 | 0 | 4 | | 4 | 2 | 6 |

We observe that:

For even n, S_n n times 2. For odd n, S_n (n - 1) times 2 2.

Expressed in a compact form:

S_n n times frac{3 - (-1)^n}{2}

Applying the limit, we get:

[lim_{n to infty} S_n infty]

This confirms the divergence of the series to infinity.

Conclusion

In conclusion, the infinite series S sum_{n0}^{infty} 1 - (-1)^n diverges to infinity. This example highlights the importance of careful analysis and the application of convergence tests in understanding the behavior of infinite series.