[tex]\sum_{n=0}^a 2^n(z-i)^{4 n}[/tex]
Answer (1)
Rewrite the sum as a geometric series: ∑ n = 0 a ( 2 ( z − i ) 4 ) n .
Apply the formula for the sum of a finite geometric series: 1 − r 1 − r a + 1 , where r = 2 ( z − i ) 4 .
The sum is 1 − 2 ( z − i ) 4 1 − ( 2 ( z − i ) 4 ) a + 1 if 2 ( z − i ) 4 = 1 , and a + 1 if 2 ( z − i ) 4 = 1 .
The final answer is expressed as a piecewise function: { 1 − 2 ( z − i ) 4 1 − ( 2 ( z − i ) 4 ) a + 1 a + 1 if 2 ( z − i ) 4 = 1 if 2 ( z − i ) 4 = 1
Explanation
Understanding the Problem We are given the sum ∑ n = 0 a 2 n ( z − i ) 4 n . Our goal is to find a closed-form expression for this sum.
Rewriting the Sum We can rewrite the summand as 2 n ( z − i ) 4 n = 2 n (( z − i ) 4 ) n = ( 2 ( z − i ) 4 ) n . Thus, the sum becomes ∑ n = 0 a ( 2 ( z − i ) 4 ) n .
Recognizing Geometric Series This is a geometric series with the first term 1 (when n = 0 ) and common ratio r = 2 ( z − i ) 4 . The formula for the sum of a finite geometric series is ∑ n = 0 a r n = 1 − r 1 − r a + 1 , where r = 1 .
Applying the Formula Substituting r = 2 ( z − i ) 4 into the formula, we get the closed-form expression: 1 − 2 ( z − i ) 4 1 − ( 2 ( z − i ) 4 ) a + 1 .This holds when 2 ( z − i ) 4 = 1 .
Considering the Special Case If 2 ( z − i ) 4 = 1 , then the sum is simply ∑ n = 0 a 1 = a + 1 .
Final Answer Therefore, the sum can be expressed as a piecewise function: n = 0 ∑ a 2 n ( z − i ) 4 n = { 1 − 2 ( z − i ) 4 1 − ( 2 ( z − i ) 4 ) a + 1 a + 1 if 2 ( z − i ) 4 = 1 if 2 ( z − i ) 4 = 1
Examples
Geometric series appear in many areas of mathematics and physics. For example, they are used in calculating compound interest, probabilities, and in the analysis of electrical circuits. Understanding how to find the sum of a geometric series allows us to solve problems involving exponential growth or decay, such as the spread of a disease or the depreciation of an asset.
