what is this equal to sum_(n=1)^(oo)(1)/(n)=1+(1)/(2)+(1)/(3)+cdots

Recognize Nature of Series: The series sum from n=1 to infinity of 1/n is known as the harmonic series. To determine its value, we need to recognize the nature of this series.Harmonic Series Divergence: The harmonic series is a well-known divergent series. This means that as n approaches infinity, the sum of the series does not converge to a finite limit; instead, it grows without bound.Comparison with Integral: To illustrate the divergence, we can compare the harmonic series to the integral of 1/x from 1 to infinity. The integral of 1/x dx from 1 to infinity is equal to the natural logarithm of x evaluated from 1 to infinity, which is infinite. Since the integral is a lower bound for the sum of the series, this implies that the harmonic series also diverges to infinity.Series Sum Divergence: Therefore, the sum of the series sum from n=1 to infinity of 1/n does not equal a finite number. Instead, it is said to be divergent or infinite.

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what is this equal to
sum_(n=1)^(oo)(1)/(n)=1+(1)/(2)+(1)/(3)+cdots

what is this equal to $\newline$ $\sum_{n=1}^{\infty}\frac{1}{n}=1+\frac{1}{2}+\frac{1}{3}+\cdots$

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Algebra 2

Sum of finite series starts from 1

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Q. what is this equal to

\newline

\sum_{n=1}^{\infty}\frac{1}{n}=1+\frac{1}{2}+\frac{1}{3}+\cdots

Recognize Nature of Series: The series sum from $n=1$ to $\infty$ of $\frac{1}{n}$ is known as the harmonic series. To determine its value, we need to recognize the nature of this series.
Harmonic Series Divergence: The harmonic series is a well-known divergent series. This means that as $n$ approaches infinity, the sum of the series does not converge to a finite limit; instead, it grows without bound.
Comparison with Integral: To illustrate the divergence, we can compare the harmonic series to the integral of $\frac{1}{x}$ from $1$ to infinity. The integral of $\frac{1}{x} \, dx$ from $1$ to infinity is equal to the natural logarithm of $x$ evaluated from $1$ to infinity, which is infinite. Since the integral is a lower bound for the sum of the series, this implies that the harmonic series also diverges to infinity.
Series Sum Divergence: Therefore, the sum of the series sum from $n=1$ to $\infty$ of $\frac{1}{n}$ does not equal a finite number. Instead, it is said to be divergent or infinite.