Evaluate the series shown. Then write an equivalent series using summation notation such that the lower index starts at 0.
[tex]$\sum_{n=1}^3 2(n+5)$[/tex]
Answer (1)
Evaluate the series by expanding the summation: ∑ n = 1 3 2 ( n + 5 ) = 2 ( 6 ) + 2 ( 7 ) + 2 ( 8 ) = 42 .
Adjust the index variable: Let m = n − 1 , so n = m + 1 .
Substitute n = m + 1 into the expression 2 ( n + 5 ) to get 2 ( m + 6 ) .
Rewrite the series: ∑ n = 0 2 2 ( n + 6 ) . The evaluated series is 42, and the equivalent series with the lower index starting at 0 is n = 0 ∑ 2 2 ( n + 6 ) .
Explanation
Understanding the Problem We are given the series ∑ n = 1 3 2 ( n + 5 ) and asked to evaluate it and then rewrite it using summation notation such that the lower index starts at 0.
Evaluating the Series First, let's evaluate the series by expanding the summation: n = 1 ∑ 3 2 ( n + 5 ) = 2 ( 1 + 5 ) + 2 ( 2 + 5 ) + 2 ( 3 + 5 ) = 2 ( 6 ) + 2 ( 7 ) + 2 ( 8 ) = 12 + 14 + 16 = 42 So, the sum of the series is 42.
Rewriting the Series Now, let's rewrite the series with the lower index starting at 0. To do this, we need to adjust the index variable. Let m = n − 1 , so n = m + 1 . When n = 1 , m = 0 . When n = 3 , m = 2 . Thus, m ranges from 0 to 2. Substitute n = m + 1 into the expression 2 ( n + 5 ) to get: 2 (( m + 1 ) + 5 ) = 2 ( m + 6 ) So, the series can be rewritten as: m = 0 ∑ 2 2 ( m + 6 ) Replacing m with n , we have: n = 0 ∑ 2 2 ( n + 6 ) Let's verify that the new series is equivalent to the original series: n = 0 ∑ 2 2 ( n + 6 ) = 2 ( 0 + 6 ) + 2 ( 1 + 6 ) + 2 ( 2 + 6 ) = 2 ( 6 ) + 2 ( 7 ) + 2 ( 8 ) = 12 + 14 + 16 = 42 Since the sum is the same, the new series is equivalent to the original series.
Final Answer Therefore, the evaluated series is 42, and the equivalent series with the lower index starting at 0 is ∑ n = 0 2 2 ( n + 6 ) .
Examples
Understanding series and summation notation is crucial in many fields, such as physics and engineering. For example, when calculating the total energy of a system with multiple components, you might use a series to sum the energy contributions from each component. Similarly, in signal processing, Fourier series are used to decompose complex signals into simpler sinusoidal components. By manipulating the summation indices, engineers can simplify calculations and optimize system performance.
