Mathematical Series Types

Arithmetic Series

An arithmetic series is a sequence of numbers in which each term increases by a constant difference, called the common difference. For example, if the first term of the series is 2 and the common difference is 3, the sequence becomes: 2, 5, 8, 11, and so on. The series is formed by summing the terms of this sequence. For instance, the sum of the first five terms of the above sequence is:$2 + 5 + 8 + 11 + 14 = 40$.

Formula for Arithmetic Series

• Sum of the first $n$ terms:
  $$S_n = \sum_{k=1}^{n} (a + (k-1)d)$$

  Here,

  • $a$ is the first term
  • $d$ is the common difference
  • $n$ is the number of terms
• Alternative formula for the sum:
  $$S_n = \frac{n}{2} (2a + (n-1)d)$$

  This formula is useful when the first term and common difference are known.

• Sum of terms when the last term is known:
  $$S_n = \frac{n}{2} (a + l)$$

  Here, $l$ represents the last term of the series.

Summing Terms with a Constant Difference

When terms of a series increase or decrease by a constant difference, the sum can be computed using the above formulas. For instance, the sum of the first six terms of the series 3, 7, 11, 15, 19, 23 is:

Infinite Arithmetic Series

Unlike geometric series, an arithmetic series does not converge unless the common difference is zero. For example, the series 2, 5, 8, 11, ... grows indefinitely as more terms are added.

Advanced Questions

Question 1: Finding the Number of Terms

If the sum of an arithmetic series is 210, the first term is 5, and the common difference is 3, find the number of terms.

Solution:

Substitute $S_n = 210, a = 5, d = 3$:

Using the quadratic formula:

Thus, $n = 12$ (since $n > 0$).

Question 2: Infinite Series with Constant Difference of Zero

What is the sum of the series where all terms are equal, such as 5, 5, 5, ...?

Solution:

When the common difference is zero, the series becomes a constant value repeated. The sum can be written as:

For infinite terms, this sum diverges unless additional constraints are applied, such as summing over a finite interval.

Geometric Series

A geometric series is a sequence of numbers in which each term is obtained by multiplying the previous term by a constant ratio, called the common ratio. For example, if the first term is 2 and the common ratio is 3, the sequence becomes: 2, 6, 18, 54, and so on. The series is formed by summing the terms of this sequence. For instance, the sum of the first four terms of the sequence is:$2 + 6 + 18 + 54 = 80$.

Formulas for Geometric Series

• Sum of the first $n$ terms:
  $$S_n = \sum_{k=0}^{n-1} ar^k$$

  This can be simplified to:

  $$S_n = a \frac{1 - r^n}{1 - r}, \quad \text{if } r \neq 1$$

  Here,

  • $a$ is the first term
  • $r$ is the common ratio
  • $n$ is the number of terms
• Sum of an infinite geometric series:
  $$S_{\infty} = \frac{a}{1-r}, \quad \text{if } |r| < 1$$

  This formula is valid only when the series converges, i.e., $|r| < 1$.

Examples

Example 1: Finite Geometric Series

Find the sum of the first 5 terms of the series: 3, 6, 12, 24, 48.

Solution:

Here, $a = 3, r = 2, n = 5$. Using the formula:

So, the sum is 93.

Example 2: Infinite Geometric Series

Find the sum of the series: 4, 2, 1, 0.5, ..., up to infinity.

Solution:

Here, $a = 4, r = 0.5$. Since $|r| < 1$, the series converges. Using the formula:

So, the sum is 8.

Advanced Questions

Question 1: Finding the Number of Terms

If the sum of a geometric series is 728, the first term is 2, and the common ratio is 3, find the number of terms.

Solution:

Substitute $S_n = 728, a = 2, r = 3$:

Taking the logarithm:

So, the series has 6 terms.

Question 2: Convergence of an Infinite Series

Determine whether the series 5, -2.5, 1.25, -0.625, ... converges, and if so, find its sum.

Solution:

Here, $a = 5, r = -0.5$. Since $|r| = 0.5 < 1$, the series converges. Using the formula:

So, the sum is $\frac{10}{3}$.

Harmonic Series

The harmonic series is a series formed by the sum of the reciprocals of natural numbers. It is represented as:

While the terms of the harmonic series decrease as they progress, the series itself grows without bound and diverges to infinity. However, the rate of growth is very slow, much slower than that of other divergent series like the arithmetic series.

Properties of the Harmonic Series

• Divergence: The harmonic series does not converge but grows unbounded, as proved using the comparison test with a divergent series.
• Growth Rate: The sum of the first $n$ terms, $H_n$, grows approximately as $\ln(n) + \gamma$, where $\gamma$ is the Euler-Mascheroni constant.
• Applications: The harmonic series frequently appears in mathematics, physics, and computer science, particularly in contexts involving logarithmic growth or complexity analysis (e.g., sorting algorithms).

Generalized Harmonic Series

The generalized harmonic series is an extension where each term is raised to a power $p$:

This series converges if and only if $p > 1$, as the terms decrease rapidly enough to produce a finite sum.

Examples

Example 1: Partial Sum

Calculate the sum of the first 5 terms of the harmonic series.

Solution:

The sum is approximately $2.2833$.

Example 2: Generalized Harmonic Series

Determine if the series $H_p$ with $p = 2$ converges, and if so, find its approximate sum.

Solution:

For $p = 2$, the series becomes:

This is the famous Basel problem, and its sum is:

Since $p > 1$, the series converges.

Advanced Questions

Question 1: Divergence Test

Prove that the harmonic series diverges.

Solution:

Use the comparison test. Group the terms as follows:

Observe that each group is greater than or equal to $\frac{1}{2}$, leading to an unbounded sum. Thus, the harmonic series diverges.

Question 2: Approximation for Large $n$

Estimate $H_{1000}$ using the approximation $H_n \approx \ln(n) + \gamma$, where $\gamma \approx 0.577$.

Solution:

So, $H_{1000} \approx 7.4858$.

P-Series

A p-series is a special type of infinite series where each term is the reciprocal of a natural number raised to the power of $p$. It is mathematically expressed as:

The behavior of a p-series depends entirely on the value of $p$:

• Convergence: The series converges if $p > 1$, as the terms decrease rapidly enough to produce a finite sum.
• Divergence: The series diverges if $p \leq 1$, as the terms do not diminish quickly enough to prevent an infinite sum.

Key Properties

• Comparison with Harmonic Series: When $p = 1$, the p-series reduces to the harmonic series, which is divergent.
• Rate of Convergence: As $p$ increases, the series converges more quickly, since the terms diminish at a faster rate.
• Special Case: For $p = 2$, the p-series becomes the Basel problem, where the sum is $\frac{\pi^2}{6}$.

Examples

Example 1: Convergent p-series

Evaluate the p-series with $p = 2$:

Solution:

This is the Basel problem, and its sum is:

Example 2: Divergent p-series

Determine if the series $\sum_{n=1}^{\infty} \frac{1}{n}$ converges or diverges.

Solution:

Here, $p = 1$, so the series reduces to the harmonic series, which diverges. The terms do not decrease quickly enough to yield a finite sum.

Advanced Questions

Question 1: Convergence Test for a p-series

Show that the p-series converges for $p > 1$.

Solution:

Apply the integral test. The function $f(x) = \frac{1}{x^p}$ is positive, continuous, and decreasing for $x \geq 1$. Compute the improper integral:

For $p > 1$:

The integral converges, and so does the p-series.

Question 2: Partial Sum Approximation

Approximate the sum of the first 10 terms of the p-series with $p = 3$.

Solution:

Using a calculator, the sum is approximately:

Question 3: Divergence for $p \leq 1$

Prove that the p-series diverges for $p \leq 1$.

Solution:

For $p \leq 1$, the terms $\frac{1}{n^p}$ do not decrease rapidly enough. Specifically, for $p = 1$, the terms are identical to the harmonic series, which diverges. For $p < 1$, the terms decrease even more slowly, ensuring divergence.

Alternating Series

An alternating series is a series where the terms alternate in sign. Such series are represented as:

For example:

Alternating series often converge under certain conditions, even if the corresponding non-alternating series diverges.

Convergence Test

The **Alternating Series Test** (Leibniz Criterion) states that an alternating series converges if the following conditions are met:

• The absolute values of the terms decrease monotonically: $|a_{n+1}| < |a_n|$
• The terms approach zero: $\lim_{n \to \infty} a_n = 0$

If both conditions are satisfied, the series converges.

Absolute and Conditional Convergence

• Absolute Convergence: The series converges absolutely if the series of absolute values, $\sum |a_n|$, converges.
• Conditional Convergence: If the series converges but does not converge absolutely, it is said to converge conditionally.

Examples

Example 1: Alternating Harmonic Series

Determine if the series $S = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \dots$ converges.

Solution:

The absolute values of terms form the harmonic series, which diverges. However, this is an alternating series, and:

• The terms decrease monotonically: $\frac{1}{n+1} < \frac{1}{n}$
• The terms approach zero: $\lim_{n \to \infty} \frac{1}{n} = 0$

By the Alternating Series Test, the series converges. This is an example of conditional convergence.

Example 2: Non-Converging Alternating Series

Does the series $S = 1 - 1 + 1 - 1 + \dots$ converge?

Solution:

Although the terms alternate, they do not approach zero: $\lim_{n \to \infty} a_n \neq 0$. Therefore, the series diverges.

Advanced Questions

Question 1: Convergence of a General Alternating Series

Prove that the series $\sum_{n=1}^{\infty} \frac{(-1)^n}{n^2}$ converges.

Solution:

The series alternates in sign, and the absolute terms are $\frac{1}{n^2}$. These terms:

• Decrease monotonically: $\frac{1}{(n+1)^2} < \frac{1}{n^2}$
• Approach zero: $\lim_{n \to \infty} \frac{1}{n^2} = 0$

By the Alternating Series Test, the series converges. Additionally, since $\sum \frac{1}{n^2}$ converges, this series converges absolutely.

Question 2: Estimation of Partial Sum

Approximate the sum of the first four terms of the alternating harmonic series.

Solution:

The first four terms are:

Calculating:

The partial sum is approximately $0.583$.

Question 3: Remainder Estimation

Show that the remainder after the first four terms of the alternating harmonic series is less than the fifth term.

Solution:

For an alternating series, the error in approximating the sum using $n$ terms is less than the magnitude of the first omitted term. Here:

Thus, the remainder is less than $0.2$.

Telescoping Series

A telescoping series is a series in which most terms cancel out due to the structure of the terms, simplifying the calculation of the sum.

The cancellation occurs because each negative term in one factor cancels with the positive term of the next, leaving only the first and last terms.

General Formula

For a telescoping series:

After cancellation:

For an infinite telescoping series, if $\lim_{n \to \infty} a_{n+1} = 0$, the sum is:

Example

Consider the series:

Most terms cancel, leaving:

Since $\lim_{n \to \infty} \frac{1}{n+1} = 0$, the sum is:

Advanced Examples

Example 1: Telescoping Rational Series

Find the sum of the series:

Solution:

The series is telescoping. After cancellation:

Taking the limit as $n \to \infty$:

The sum of the series is $1$.

Example 2: Finite Telescoping Series

Compute the sum of the finite series:

Solution:

Write out the terms:

Most terms cancel, leaving:

The sum of the series is $\frac{4}{5}$.

Advanced Questions

Question 1: Infinite Telescoping Series

Prove that the series $\sum_{n=1}^{\infty} \frac{1}{n(n+1)}$ converges and find its sum.

Solution:

Rewrite the general term using partial fractions:

The series becomes:

This is a telescoping series. After cancellation, the sum is:

Question 2: Non-Trivial Telescoping Series

Compute the sum of the series:

Solution:

The series is telescoping. After cancellation:

Taking the limit as $n \to \infty$:

The sum of the series is $\frac{1}{4}$.