Question

Prove Using The Monotone Convergence Theorem That The Infinite Series ∑N=1[infinity]N21 Converges.

Answer 1

To prove that the infinite series ∑N=1∞ N^2 converges using the Monotone Convergence Theorem, we need to show that it is bounded and monotonically increasing.

Step-by-step explanation:

In order to prove that the infinite series ∑N=1∞ N^2 converges using the Monotone Convergence Theorem, we need to show that it is bounded and monotonically increasing.

1. Boundedness: We can see that every term in the series N^2 is non-negative, so the series is also non-negative. Additionally, we know that N^2 is always greater than or equal to N. Therefore, the series is greater than or equal to the series ∑N=1∞ N, which is a well-known divergent series. Thus, the series ∑N=1∞ N^2 is also divergent.
2. Monotonicity: We can observe that each term in the series N^2 is greater than or equal to the previous term. Therefore, the series is monotonically increasing.

Since the infinite series ∑N=1∞ N^2 is both bounded and monotonically increasing, we can conclude that it converges by the Monotone Convergence Theorem.