Question

Suppose that \(\sum_{n=1}^{\infty} a_{n}=1\), that \(\sum_{n=1}^{\infty} b_{n}=-1\), that \(a_{1}=2\), and \(b_{1}=-3 .\) Find the sum of the indicated series. $$ \sum_{n=2}^{\infty}\left(a_{n}-b_{n}\right) $$

Short answer

The sum is -3.

Step-by-step solution

Understand the Problem

We are given two series, namely, \(\sum_{n=1}^\infty a_n = 1\) and \(\sum_{n=1}^\infty b_n = -1\). Additionally, it's given that \(a_1 = 2\) and \(b_1 = -3\). We need to find the sum of \(\sum_{n=2}^{\infty}(a_n - b_n)\). The provided series starts from \(n = 2\), so we must first exclude the term when \(n = 1\).

Calculate the Sum to Exclude the First Term for Series \(a_n\)

Given \(\sum_{n=1}^{\infty} a_n = 1\) and \(a_1 = 2\), we calculate \(\sum_{n=2}^{\infty} a_n = 1 - a_1 = 1 - 2 = -1\).

Calculate the Sum to Exclude the First Term for Series \(b_n\)

Given \(\sum_{n=1}^{\infty} b_n = -1\) and \(b_1 = -3\), we calculate \(\sum_{n=2}^{\infty} b_n = -1 - b_1 = -1 + 3 = 2\).

Combine the Results

We now need to calculate \(\sum_{n=2}^{\infty} (a_n - b_n)\) which is equivalent to \(\sum_{n=2}^{\infty} a_n - \sum_{n=2}^{\infty} b_n\). Substitute the previously calculated sums: \(-1 - 2 = -3\).

Key concepts

Convergence

Understanding convergence is crucial when dealing with infinite series. Convergence, in the context of an infinite series, refers to the behavior of the series as it extends towards infinity. A series is said to converge if the sum of its terms approaches a finite limit.
- When mathematically defining convergence, we look at the series \(\sum_{n=1}^{\infty} a_n\) and ask: Does this sum approach a specific value as \(n\) increases?- For example, if \(\sum_{n=1}^{\infty} a_n = 1\), it means that the series converges to 1.- Similarly, \(\sum_{n=1}^{\infty} b_n = -1\) means the series converges to -1.By understanding the limits to which these series converge, we can further explore operations on them, such as combining series or removing particular terms. Convergence ensures that the operations on the series lead to meaningful mathematical conclusions.

Series Subtraction

Series subtraction involves finding the difference between two series. This means calculating the series \(\sum_{n=1}^{\infty} (a_n - b_n)\).
- Essentially, series subtraction allows us to subtract every corresponding term in one series from another.- Consider two convergent series, \(\sum_{n=1}^\infty a_n\) and \(\sum_{n=1}^\infty b_n\). The series \(\sum_{n=1}^\infty (a_n - b_n)\) will converge under certain conditions.- For instance, if both series converge individually as given, the series formed by their subtraction will also converge to the subtraction of their respective sums.This operation is instrumental when calculating combinations of infinite series, allowing us to leverage known sums to find unknowns, as demonstrated in the exercise problem.

Exclusion of Terms

When working with series, it is often necessary to exclude certain terms, especially terms that may not solve directly under given constraints.
- The exclusion of terms technique involves adjusting the sum of the series to start at an index different from the initial one.- In the given problem, \ it means ignoring the first terms, \(a_1\) and \(b_1\), and considering only terms starting from \(n=2\).- This process is handled by subtracting the first term: for \(\sum_{n=1}^{\infty} a_n = 1\) and \(a_1=2\), this results in \(\sum_{n=2}^{\infty} a_n = 1 - 2 = -1\).Similarly for \(b_n\), knowing \(\sum_{n=1}^{\infty} b_n = -1\) and \(b_1 = -3\), the sum from \(n=2\) onwards is corrected to \(2\). This technique is fundamental for accurately finding sums when manipulating only parts of the original series.