What is the sum of the series sqrt5-(5)/(2)+(5sqrt5)/(3)-(25)/(4)+dots+(-1)^(n)(sqrt5)/(n+1)+dots? (A) ln(1+sqrt5) (B) e_(sqrt5) (C) ln(sqrt5) (D) sqrt5 (E) The series diverges.

Simplify General Term: Simplify the general term of the series: (-1)^n(sqrt5)/(n+1). Recognize Alternating Series: Recognize the series as an alternating series with a decreasing sequence sqrt5/(n+1). Apply Sum Formula: Apply the formula for the sum of an alternating harmonic series with a constant multiple: ln(1 + sqrt5) * sqrt5.

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What is the sum of the series
sqrt5-(5)/(2)+(5sqrt5)/(3)-(25)/(4)+dots+(-1)^(n)(sqrt5)/(n+1)+dots?
(A)
ln(1+sqrt5)
(B)
e_(sqrt5)
(C)
ln(sqrt5)
(D)
sqrt5
(E) The series diverges.

What is the sum of the series $\newline$ $\sqrt{5}-\frac{5}{2}+\frac{5\sqrt{5}}{3}-\frac{25}{4}+\dots+(-1)^{n}\frac{\sqrt{5}}{n+1}+\dots$ ? $\newline$ (A) $\ln(1+\sqrt{5})$ $\newline$ (B) $e_{\sqrt{5}}$ $\newline$ (C) $\ln(\sqrt{5})$ $\newline$ (D) $\sqrt{5}$ $\newline$ (E) The series diverges.

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Math Problems

Algebra 2

Sum of finite series starts from 1

Full solution

Q. What is the sum of the series

\newline

\sqrt{5}-\frac{5}{2}+\frac{5\sqrt{5}}{3}-\frac{25}{4}+\dots+(-1)^{n}\frac{\sqrt{5}}{n+1}+\dots

\newline

(A)

\ln(1+\sqrt{5})

\newline

(B)

e_{\sqrt{5}}

\newline

(C)

\ln(\sqrt{5})

\newline

(D)

\sqrt{5}

\newline

(E) The series diverges.

Simplify General Term: Simplify the general term of the series: $(-1)^n\sqrt{5}/(n+1)$ .
Recognize Alternating Series: Recognize the series as an alternating series with a decreasing sequence $\sqrt{5}/(n+1)$ .
Apply Sum Formula: Apply the formula for the sum of an alternating harmonic series with a constant multiple: $\ln(1 + \sqrt{5}) \times \sqrt{5}$ .