Final answer:
To prove convergence with Bernoulli's inequality, we need to show that a series is bounded above by another series which converges.
We can use Bernoulli's inequality to find a series that bounds the given series and converges. For example, the series ∑(1/n^2) is bounded above by the convergent geometric series ∑(1/2^n).
Step-by-step explanation:
To prove convergence with Bernoulli's inequality, we need to show that a series is bounded above by another series which converges.
Bernoulli's inequality states that for any real number x ≥ -1 and any positive integer n, (1 + x)^n ≥ 1 + nx.
Let's suppose we have a series ∑a_n. If we can find a number r such that |a_n| ≤ r^n for all n and the series ∑r^n converges, then we can conclude that the series ∑a_n also converges.
This is because the series can be bounded above by a convergent geometric series with a common ratio between -1 and 1.
For example, consider the series ∑(1/n^2). We can bound this series above by ∑(1/2^n), as for all n, (1/2^n) ≤ (1/n^2). Since ∑(1/2^n) is a convergent geometric series, we can conclude that the series ∑(1/n^2) also converges.