Evaluate. sum_(n=0)^(oo)n!(cos ((1)/(sqrt(n!)))-1)

Find First Term: The series starts at n=0, so let's find the first term. For n=0, the term is 0!(cos(1/sqrt(0!))-1) which is 1(cos(1/1)-1) = 0.General Term for n>0: Now, let's consider the general term for n>0. We have n!(cos(1/sqrt(n!))-1). Since cos(1) is close to 0.54 and decreases as the argument gets smaller, the term cos(1/sqrt(n!)) will approach 1 as n increases.Approaching 1: Because cos(1/sqrt(n!)) approaches 1, the expression (cos(1/sqrt(n!))-1) approaches 0. This means that for large n, the terms of the series become very small.Factorial Growth vs. Cosine: However, we are multiplying by n!, which grows very fast. So we need to check if the factorial growth or the cosine approaching 1 has a stronger effect on the terms.Factorial Dominance: For large n, n! grows faster than any polynomial function, and since the cosine function is bounded between -1 and 1, the factorial will dominate. This means the terms will not approach 0, and the series will diverge.Series Divergence: Since the series diverges, it does not have a finite sum. Therefore, we cannot calculate the sum of the series as it does not converge to a specific value.

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Evaluate.
sum_(n=0)^(oo)n!(cos ((1)/(sqrt(n!)))-1)

Evaluate. $\newline$ $\sum_{n=0}^{\infty} n !\left(\cos \frac{1}{\sqrt{n !}}-1\right)$

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Algebra 2

Sum of finite series starts from 1

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Q. Evaluate.

\newline

\sum_{n=0}^{\infty} n !\left(\cos \frac{1}{\sqrt{n !}}-1\right)

Find First Term: The series starts at $n=0$ , so let's find the first term. For $n=0$ , the term is $0!(\cos(1/\sqrt{0!})-1)$ which is $1(\cos(1/1)-1) = 0$ .
General Term for n>0: Now, let's consider the general term for n>0. We have $n!(\cos(\frac{1}{\sqrt{n!}})-1)$ . Since $\cos(1)$ is close to $0.54$ and decreases as the argument gets smaller, the term $\cos(\frac{1}{\sqrt{n!}})$ will approach $1$ as $n$ increases.
Approaching $1$ : Because $\cos(\frac{1}{\sqrt{n!}})$ approaches $1$ , the expression $\left(\cos\left(\frac{1}{\sqrt{n!}}\right)-1\right)$ approaches $0$ . This means that for large $n$ , the terms of the series become very small.
Factorial Growth vs. Cosine: However, we are multiplying by $n!$ , which grows very fast. So we need to check if the factorial growth or the cosine approaching $1$ has a stronger effect on the terms.
Factorial Dominance: For large $n$ , $n!$ grows faster than any polynomial function, and since the cosine function is bounded between $-1$ and $1$ , the factorial will dominate. This means the terms will not approach $0$ , and the series will diverge.
Series Divergence: Since the series diverges, it does not have a finite sum. Therefore, we cannot calculate the sum of the series as it does not converge to a specific value.