Problem 39
Question
If the \(n\) th partial sum of a series \(\sum_{n=1}^{\infty} a_{n}\) is $$s_{n}=\frac{n-1}{n+1}$$ find \(a_{n}\) and \(\sum_{n=1}^{\infty} a_{n}\)
Step-by-Step Solution
Verified Answer
The series diverges, as \( a_n = \frac{2}{n} \) and \( \sum_{n=1}^{\infty} \frac{1}{n} \) diverges.
1Step 1: Define the Partial Sum
The partial sum of the series is given by \( s_n = \frac{n - 1}{n + 1} \). This represents the sum of the first \( n \) terms of the series \( \sum_{n=1}^{\infty} a_n \).
2Step 2: Determine Expression for a_n
To find \( a_n \), use the relationship \( a_n = s_n - s_{n-1} \). First, express \( s_{n-1} \): \[s_{n-1} = \frac{(n-1) - 1}{(n-1) + 1} = \frac{n-2}{n} \]Now, calculate \( a_n \):\[a_n = s_n - s_{n-1} = \frac{n-1}{n+1} - \frac{n-2}{n} \]
3Step 3: Simplify a_n
Simplify \( a_n \) by finding a common denominator for the fractions \( \frac{n-1}{n+1} \) and \( \frac{n-2}{n} \):\[a_n = \frac{(n-1)n - (n-2)(n+1)}{(n+1)n} \]Simplifying the numerator:\[(n-1)n - (n-2)(n+1) = n^2 - n - (n^2 - n - 2n - 2) = (n^2 - n) - (n^2 - 3n - 2) = 2n + 2\]So,\[ a_n = \frac{2(n + 1)}{n(n+1)} = \frac{2}{n} \]
4Step 4: Find the Sum of the Infinite Series
Since \( a_n = \frac{2}{n} \), the infinite series is \( \sum_{n=1}^{\infty} a_n = 2\sum_{n=1}^{\infty} \frac{1}{n} \). The series \( \sum_{n=1}^{\infty} \frac{1}{n} \) is a harmonic series, which is known to diverge. Therefore, the series \( \sum_{n=1}^{\infty} a_n \) also diverges.
Key Concepts
Series ConvergenceHarmonic SeriesInfinite Series
Series Convergence
In mathematics, the concept of series convergence is vital when dealing with infinite series. A series is said to converge if the sequence of its partial sums approaches a specific finite number as more terms are added. In simpler terms, as you add more and more terms of the series, the total approaches a number that doesn't change. However, not all series have this property.
To determine if a series converges, we examine its partial sums. For the series \(\sum_{n=1}^{\infty} a_n\), the partial sum \(s_n\) is the sum of the first \(n\) terms. If \(s_n\) converges to a limit \(L\), then we say the series converges to \(L\).
Understanding convergence helps us decide whether a series has a finite total or not, making it extremely useful in both theoretical and applied mathematics, such as in solving problems related to physics or finance.
To determine if a series converges, we examine its partial sums. For the series \(\sum_{n=1}^{\infty} a_n\), the partial sum \(s_n\) is the sum of the first \(n\) terms. If \(s_n\) converges to a limit \(L\), then we say the series converges to \(L\).
Understanding convergence helps us decide whether a series has a finite total or not, making it extremely useful in both theoretical and applied mathematics, such as in solving problems related to physics or finance.
Harmonic Series
The harmonic series is one of the most famous examples of an infinite series. It takes the form \(\sum_{n=1}^{\infty} \frac{1}{n}\). Despite its simple appearance, this series is very special because it is known to diverge.
Divergence means that the partial sums of the harmonic series will keep on increasing without bound as more terms are added, never settling at a finite number. While individual terms \(\frac{1}{n}\) become smaller as \(n\) increases, the cumulative total continues to grow indefinitely.
The harmonic series showcases an important point about infinite series: even if the terms become very small, the entire series might still diverge. This is why careful analysis of each series is crucial to determine its convergence or divergence.
Divergence means that the partial sums of the harmonic series will keep on increasing without bound as more terms are added, never settling at a finite number. While individual terms \(\frac{1}{n}\) become smaller as \(n\) increases, the cumulative total continues to grow indefinitely.
The harmonic series showcases an important point about infinite series: even if the terms become very small, the entire series might still diverge. This is why careful analysis of each series is crucial to determine its convergence or divergence.
Infinite Series
An infinite series is essentially a sum of an infinite sequence of numbers. It differs from finite series, which have a definite number of terms. Infinite series can either converge to a finite sum or diverge, as we have seen with the harmonic series.
The sum of an infinite series is computed using its partial sums \(s_n\). If the sequence of partial sums tend towards a finite limit, then the series converges, and this limit is considered the sum of the series. Describing the behavior of infinite series is a major part of mathematical analysis.
Important concepts to remember about infinite series:
The sum of an infinite series is computed using its partial sums \(s_n\). If the sequence of partial sums tend towards a finite limit, then the series converges, and this limit is considered the sum of the series. Describing the behavior of infinite series is a major part of mathematical analysis.
Important concepts to remember about infinite series:
- Convergence: Does the series approach a specific value?
- Divergence: Do the terms or partial sums continue growing indefinitely?
- Testing Methods: Various tests (like the ratio test or integral test) examine if an infinite series converges or diverges.
Other exercises in this chapter
Problem 38
Let \(f_{n}(x)=(\sin n x) / n^{2}\) . Show that the series \(\Sigma f_{n}(x)\) converges for all values of \(x\) but the series of derivatives \(\Sigma f_{n}^{\
View solution Problem 39
Let $$ f(x)=\sum_{n=1}^{\infty} \frac{x^{n}}{n^{2}} $$ Find the intervals of convergence for \(f, f^{\prime},\) and \(f^{\prime \prime}\)
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The meaning of the decimal representation of a number $$0 . d_{1} d_{2} d_{3} \ldots\( (where the digit \)d_{i}\( is one of the numbers \)0,1\( \)2, \ldots, 9 )
View solution Problem 39
Use the Maclaurin series for cos \(x\) to compute \(\cos 5^{\circ}\) correct to five decimal places.
View solution