Linearity Properties of Series
- Addition of Series:
Formula: \( \sum_{n=1}^{\infty} (a_n + b_n) = \sum_{n=1}^{\infty} a_n + \sum_{n=1}^{\infty} b_n \)
Explanation: You can add two series together by adding their corresponding terms. This property allows the addition operation to be distributed over the sum.
- Subtraction of Series:
Formula: \( \sum_{n=1}^{\infty} (a_n – b_n) = \sum_{n=1}^{\infty} a_n – \sum_{n=1}^{\infty} b_n \)
Explanation: You can subtract one series from another by subtracting their corresponding terms. This property allows the subtraction operation to be distributed over the sum.
- Constant Multiple of Series:
Formula: \( \sum_{n=1}^{\infty} c \cdot a_n = c \cdot \sum_{n=1}^{\infty} a_n \), where \( c \) is a constant.
Explanation: You can multiply every term of a series by a constant, and this can be done by multiplying the sum of the series by that value. This property allows the scalar multiplication to be distributed over the sum.
These properties are referred to as the “linearity of summation” because they demonstrate how summation behaves linearly with respect to addition, subtraction, and scalar multiplication. They are foundational rules in working with series and have broad applications in mathematics.
Example: Addition of Two Convergent Series
Let’s consider two convergent series:
- Series A: \( \sum_{n=1}^{\infty} \frac{1}{2^n} \) (a convergent geometric series with sum \( \frac{1}{2} \div (1 – \frac{1}{2}) = 1 \))
- Series B: \( \sum_{n=1}^{\infty} \frac{1}{3^n} \) (another convergent geometric series with sum \( \frac{1}{3} \div (1 – \frac{1}{3}) = \frac{1}{2} \))
We want to find the sum of these two series:
\( \sum_{n=1}^{\infty} \left( \frac{1}{2^n} + \frac{1}{3^n} \right) \)
Using the property of addition of series, we can write this as:
\( \sum_{n=1}^{\infty} \frac{1}{2^n} + \sum_{n=1}^{\infty} \frac{1}{3^n} = 1 + \frac{1}{2} = \frac{3}{2} \)
This expression represents the sum of Series A and Series B, and the total sum is \( \frac{3}{2} \).
Note: The addition property is valid only when both series converge. In this example, both Series A and Series B are convergent geometric series, so we can apply the addition property to find their sum.
Explanation: The addition property allows us to add the corresponding terms of the two series together. In this example, we added the \( n \)-th term of Series A with the \( n \)-th term of Series B for each \( n \). This resulted in a new series that represents the sum of the original two series. The property of addition of series enables us to work with the sum of two series in a straightforward and systematic way, provided that both series converge.
Example: Difference of Two Convergent Series
Let’s consider the same two convergent series as before:
- Series A: \( \sum_{n=1}^{\infty} \frac{1}{2^n} \) (a convergent geometric series with sum \( \frac{1}{2} \div (1 – \frac{1}{2}) = 1 \))
- Series B: \( \sum_{n=1}^{\infty} \frac{1}{3^n} \) (another convergent geometric series with sum \( \frac{1}{3} \div (1 – \frac{1}{3}) = \frac{1}{2} \))
We want to find the difference between these two series:
\( \sum_{n=1}^{\infty} \left( \frac{1}{2^n} – \frac{1}{3^n} \right) \)
Using the property of subtraction of series, we can write this as:
\( \sum_{n=1}^{\infty} \frac{1}{2^n} – \sum_{n=1}^{\infty} \frac{1}{3^n} = 1 – \frac{1}{2} = \frac{1}{2} \)
This expression represents the difference between Series A and Series B, and the total difference is \( \frac{1}{2} \).
Note: The difference property is valid only when both series converge. In this example, both Series A and Series B are convergent geometric series, so we can apply the difference property to find their difference.
Explanation: The difference property allows us to subtract the corresponding terms of the two series. In this example, we subtracted the \( n \)-th term of Series B from the \( n \)-th term of Series A for each \( n \). This resulted in a new series that represents the difference between the original two series. The property of subtraction of series enables us to work with the difference of two series in a systematic way, provided that both series converge.
Example: Constant Multiple of a Convergent Series
Consider a convergent geometric series:
- Series A: \( \sum_{n=1}^{\infty} \frac{1}{2^n} \) (a convergent geometric series with sum \( \frac{1}{2} \div (1 – \frac{1}{2}) = 1 \))
We want to find the constant multiple of this series by a factor of 3:
\( 3 \cdot \sum_{n=1}^{\infty} \frac{1}{2^n} \)
Using the constant multiple property, we can distribute the constant inside the summation:
\( 3 \cdot \left( \sum_{n=1}^{\infty} \frac{1}{2^n} \right) = \sum_{n=1}^{\infty} \left( 3 \cdot \frac{1}{2^n} \right) \)
This means we multiply each term inside the summation by the constant 3:
\( \sum_{n=1}^{\infty} \left( 3 \cdot \frac{1}{2^n} \right) = 3 \cdot \frac{1}{2} + 3 \cdot \frac{1}{4} + 3 \cdot \frac{1}{8} + \ldots \)
The sum of this new series is:
\( 3 \cdot \frac{1}{2} \div (1 – \frac{1}{2}) = 3 \)
Note: The constant multiple property is valid only when the series converges. In this example, Series A is a convergent geometric series, so we can apply the constant multiple property to find the constant multiple of the series.
Explanation: The constant multiple property allows us to multiply each term of the series by a constant factor. In this example, we multiplied each term of Series A by 3. This resulted in a new series that represents the constant multiple of the original series by a factor of 3. The property of constant multiplication enables us to work with the constant multiple of a series in a systematic way, provided that the series converges.