Infinite Series Convergence

Front

What is a p-series?

Back

A p-series is a series of the form ∑(1/n^p) where p is a constant. It converges if p > 1 and diverges if p ≤ 1.

Front

When does a series ∑a_n converge?

Back

A series ∑a_n converges if the sequence of its partial sums approaches a finite limit as n approaches infinity.

Front

What does it mean if a_n → 0 as n → ∞?

Back

If a_n → 0 as n → ∞, it means that the terms of the series become arbitrarily small as n increases, but this alone does not guarantee convergence of the series.

Front

What is the necessary condition for convergence of a series?

Back

A necessary condition for the convergence of a series ∑a_n is that the terms a_n must approach 0 as n approaches infinity.

Front

What is a geometric series?

Back

A geometric series is a series of the form ∑ar^n, where a is the first term and r is the common ratio. It converges if |r| < 1.

Front

What is the formula for the sum of a convergent geometric series?

Back

The sum S of a convergent geometric series can be calculated using the formula S = a / (1 - r), where a is the first term and r is the common ratio.

Front

What is the comparison test for convergence?

Back

The comparison test states that if 0 ≤ a_n ≤ b_n for all n and ∑b_n converges, then ∑a_n also converges.

Front

What is the ratio test?

Back

The ratio test involves taking the limit L = lim (n→∞) |a_(n+1)/a_n|. If L < 1, the series converges; if L > 1, it diverges; if L = 1, the test is inconclusive.

Front

What is the root test?

Back

The root test involves taking the limit L = lim (n→∞) n√|a_n|. If L < 1, the series converges; if L > 1, it diverges; if L = 1, the test is inconclusive.

Front

What is the integral test for convergence?

Back

The integral test states that if f(x) is a positive, continuous, and decreasing function for x ≥ N and a_n = f(n), then the series ∑a_n converges if and only if the integral ∫(N to ∞) f(x) dx converges.