$\begin{array}{l}\left.\varepsilon_{n=0}^a\left[\frac{2+3 i}{5-i}\right]^n(z+i)\right)^{2 n} \\ \sum_{n=0}^a 2^n(z-i)^{4 n} \\ \varepsilon_{n=0}^a \frac{2+i n}{2^n} z^n \\ \text { Let } a_n=\frac{2+i n}{2^n} a_{n t}\end{array}$
Answer (1)
Simplify the complex number in the first summation and rewrite the summation as a geometric series.
Identify the common ratio for the first and second summations and determine the convergence conditions.
Use the ratio test to determine the convergence condition for the third summation.
Analyze the behavior of the sequence a n as n approaches infinity.
The convergence conditions for the three summations are: ∣ ( 26 7 + 26 17 i ) ( z + i ) 2 ∣ < 1 , ∣ z − i ∣ < 4 2 1 , and ∣ z ∣ < 2 . The limit of the sequence a n as n approaches infinity is 0, assuming a n t is a constant.
The analysis of the given expressions and definition is complete.
The convergence conditions for the three summations are: ∣ ( 26 7 + 26 17 i ) ( z + i ) 2 ∣ < 1 , ∣ z − i ∣ < 4 2 1 , and ∣ z ∣ < 2 . The limit of the sequence a n as n approaches infinity is 0, assuming a n t is a constant.
The analysis of the given expressions and definition is complete. Analysis Complete
Explanation
Initial Analysis We are given three expressions involving summations and a definition for a sequence a n . Let's analyze each expression to understand its properties.
Analyzing the First Summation The first expression is a summation involving a complex number and a power of ( z + i ) .
n = 0 ∑ a [ 5 − i 2 + 3 i ] n ( z + i ) 2 n First, let's simplify the complex number 5 − i 2 + 3 i :
5 − i 2 + 3 i = ( 5 − i ) ( 5 + i ) ( 2 + 3 i ) ( 5 + i ) = 25 + 1 10 + 2 i + 15 i − 3 = 26 7 + 17 i = 26 7 + 26 17 i The summation can be rewritten as: n = 0 ∑ a ( 26 7 + 26 17 i ) n ( z + i ) 2 n = n = 0 ∑ a [ ( 26 7 + 26 17 i ) ( z + i ) 2 ] n This is a geometric series with the common ratio r = ( 26 7 + 26 17 i ) ( z + i ) 2 . The series converges if ∣ r ∣ < 1 .
Analyzing the Second Summation The second expression is another summation involving a power of ( z − i ) .
n = 0 ∑ a 2 n ( z − i ) 4 n = n = 0 ∑ a [ 2 ( z − i ) 4 ] n This is a geometric series with the common ratio r = 2 ( z − i ) 4 . The series converges if ∣ r ∣ < 1 , which means ∣2 ( z − i ) 4 ∣ < 1 or ∣ ( z − i ) 4 ∣ < 2 1 or ∣ z − i ∣ < 4 2 1 .
Analyzing the Third Summation The third expression is a summation involving a complex term and a power of z .
n = 0 ∑ a 2 n 2 + in z n This is a power series. To analyze its convergence, we can use the ratio test. Let a n = 2 n 2 + in z n . Then a n a n + 1 = 2 n 2 + in z n 2 n + 1 2 + i ( n + 1 ) z n + 1 = 2 ( 2 + in ) 2 + i ( n + 1 ) z = 2∣2 + in ∣ ∣2 + i ( n + 1 ) ∣ ∣ z ∣ As n → ∞ , ∣2 + in ∣ ∣2 + i ( n + 1 ) ∣ → ∣ in ∣ ∣ in ∣ = 1 . Therefore, n → ∞ lim a n a n + 1 = 2 1 ∣ z ∣ For the series to converge, we need 2 1 ∣ z ∣ < 1 , which means ∣ z ∣ < 2 .
Analyzing the Sequence a_n We are given the definition a n = 2 n 2 + in a n t . Let's analyze this sequence. If a n t is a constant, then as n → ∞ , a n → 0 . If a n t is a function of n , we need more information to determine the behavior of a n . Let's assume a n t is a constant. Then n → ∞ lim a n = n → ∞ lim 2 n 2 + in a n t = 0 because the exponential term 2 n grows faster than the linear term n .
Conclusion Based on the analysis of the given expressions, we can determine the convergence conditions for each series and the limit of the sequence a n .
Final Answer The question is to analyze the given expressions and definition. We have analyzed the convergence of each series and the behavior of the sequence a n .
Examples
Power series and complex numbers are fundamental in electrical engineering, particularly in signal processing and circuit analysis. For example, analyzing the stability of a feedback system often involves examining the convergence of power series in the complex plane. The behavior of circuits under different frequencies can be modeled using complex impedances, and the overall system response can be determined by analyzing the convergence of related series. Understanding these concepts allows engineers to design stable and efficient systems.
