Define the remainder of an infinite series.

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Understand that an infinite series is the sum of infinitely many terms, typically written as \(\sum_{n=1}^{\infty} a_n\).

Recognize that the remainder (or tail) of an infinite series after \(N\) terms is the sum of all terms from \(N+1\) to infinity, expressed as \(R_N = \sum_{n=N+1}^{\infty} a_n\).

Interpret the remainder \(R_N\) as the difference between the total sum of the infinite series \(S\) (if it converges) and the partial sum \(S_N = \sum_{n=1}^N a_n\), so \(R_N = S - S_N\).

Note that the remainder gives an estimate of the error when approximating the infinite series by its first \(N\) terms.

Understand that analyzing the remainder is crucial for determining how well partial sums approximate the infinite series and for establishing convergence criteria.

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Infinite Series

An infinite series is the sum of infinitely many terms of a sequence. It is often written as the limit of partial sums, where each partial sum adds more terms. Understanding infinite series is essential to analyze convergence and the behavior of sums that extend indefinitely.

Partial Sums

Partial sums are the sums of the first n terms of an infinite series. They form a sequence whose limit, if it exists, defines the sum of the infinite series. Partial sums help approximate the total sum and are fundamental in defining the remainder.

Remainder of an Infinite Series

The remainder is the difference between the sum of the infinite series and a partial sum. It represents the sum of all terms not yet added and measures the error when approximating the infinite sum by a finite number of terms.