Evaluate sum_(m >= 1)^(oo)(2-(1)/(m))^(m^(2))

Analyze Behavior as m Approaches Infinity: To solve the infinite series sum_(m >= 1)^(oo)(2-(1)/(m))^(m^(2)), we need to analyze the behavior of the term (2-(1)/(m))^(m^(2)) as m approaches infinity.Approach of (2-(1/m)): As m approaches infinity, the term (1/m) approaches 0. Therefore, the expression (2-(1)/(m)) approaches 2.Consider Exponent m^2: Now, we need to consider the exponent m^2. As m approaches infinity, m^2 also approaches infinity. Thus, we are looking at the behavior of 2 raised to an infinitely large power.Behavior of 2 to Infinity: Since 2 is greater than 1, raising 2 to an infinitely large power will result in an infinitely large number. Therefore, each term of the series as m approaches infinity will approach infinity.First Term Calculation: However, we must consider the first term of the series separately, where m=1. The first term is (2-(1)/(1))^(1^(2)) = (2-1)^1 = 1^1 = 1.Series Diverges to Infinity: Since the first term is finite (equal to 1) but all subsequent terms approach infinity, the sum of the series is dominated by the infinitely large terms. Therefore, the series diverges to infinity.

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Evaluate $\sum_{m \geq 1}^{\infty}\left(2-\frac{1}{m}\right)^{m^{2}}$

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Algebra 2

Sum of finite series starts from 1

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Q. Evaluate

\sum_{m \geq 1}^{\infty}\left(2-\frac{1}{m}\right)^{m^{2}}

Analyze Behavior as $m$ Approaches Infinity: To solve the infinite series $\sum_{m \geq 1}^{\infty}(2-(1)/(m))^{m^{2}}$ , we need to analyze the behavior of the term $(2-(1)/(m))^{m^{2}}$ as $m$ approaches infinity.
Approach of $2-\left(\frac{1}{m}\right)$ : As $m$ approaches infinity, the term $\left(\frac{1}{m}\right)$ approaches $0$ . Therefore, the expression $2-\left(\frac{1}{m}\right)$ approaches $2$ .
Consider Exponent $m^2$ : Now, we need to consider the exponent $m^2$ . As $m$ approaches infinity, $m^2$ also approaches infinity. Thus, we are looking at the behavior of $2$ raised to an infinitely large power.
Behavior of $2$ to Infinity: Since $2$ is greater than $1$ , raising $2$ to an infinitely large power will result in an infinitely large number. Therefore, each term of the series as $m$ approaches infinity will approach infinity.
First Term Calculation: However, we must consider the first term of the series separately, where $m=1$ . The first term is $(2-(1)/(1))^{(1^{2})} = (2-1)^1 = 1^1 = 1$ .
Series Diverges to Infinity: Since the first term is finite (equal to $1$ ) but all subsequent terms approach infinity, the sum of the series is dominated by the infinitely large terms. Therefore, the series diverges to infinity.