Infinite Series Crash Course

Front

What is an infinite series?

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An infinite series is the sum of the terms of an infinite sequence. It can be expressed as S = a1 + a2 + a3 + ... where a1, a2, a3, ... are the terms of the sequence.

Front

What is convergence in the context of infinite series?

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Convergence refers to the property of an infinite series where the sum approaches a finite limit as more terms are added.

Front

What is divergence in the context of infinite series?

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Divergence occurs when the sum of an infinite series does not approach a finite limit as more terms are added.

Front

What is a geometric series?

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A geometric series is a series of the form S = a + ar + ar^2 + ar^3 + ... where 'a' is the first term and 'r' is the common ratio.

Front

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

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The sum S of a convergent geometric series can be calculated using the formula S = a / (1 - r), where |r| < 1.

Front

What is a telescoping series?

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A telescoping series is a series where most terms cancel out when summed, leading to a simplified expression.

Front

What is the ratio test for convergence?

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The ratio test states that for a series Σan, if lim (n→∞) |an+1/an| = L, then the series converges if L < 1, diverges if L > 1, and is inconclusive if L = 1.

Front

What is the root test for convergence?

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The root test states that for a series Σan, if lim (n→∞) n√|an| = L, then the series converges if L < 1, diverges if L > 1, and is inconclusive if L = 1.

Front

What is the integral test for convergence?

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The integral test states that if f(x) is a positive, continuous, and decreasing function, then the series Σan converges if the integral from 1 to ∞ of f(x) dx converges.

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What is a power series?

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A power series is a series of the form Σan(x - c)^n, where 'c' is a constant and an represents the coefficients.