# Harmonic Series: Is ∑(1/ k+1 ) Divergent?

• foo9008
In summary, the conversation discusses the concept of harmonic series and its application to two different series, one of which is given as ∑(1/k) and the other as ∑(1/k+1). The expert summarizer explains that while the first series is divergent, the second series can also be considered harmonic because it has the same infinite sum as the first series, with k starting from one minus 1. They suggest writing out the first few terms of both series to see how they are related.

## Homework Statement

i know that k = 0 to∞∑(1/ k) is harmonic series( we know that the sum is divergent) , how about ∑(1/ k+1 ) ?

## The Attempt at a Solution

in my opinion , it's also harmonic series , because the sum is divergent . Am i right ?

foo9008 said:

## Homework Statement

i know that k = 0 to∞∑(1/ k) is harmonic series( we know that the sum is divergent) , how about ∑(1/ k+1 ) ?

## The Attempt at a Solution

in my opinion , it's also harmonic series , because the sum is divergent . Am i right ?

Note: the series cannot be ##\sum_{k=0}^{\infty} 1/k## because the first term would be 1/0. However, starting at ##k = 1## is OK.

Do you mean that one of the series is ##\sum_{k=1}^{\infty} 1/k## and the other is ##\sum_{k=1}^{\infty}(\frac{1}{k}+1)##? That is what you wrote. Did you really mean ##\sum_{k=1}^{\infty} 1/(k+1)## for the second series? If so, use parentheses, like this: 1/(k+1).

Anyway, just write out the first first few terms of both of your series, to see how they are related.

When you say "it's also harmonic series , because the sum is divergent" you have it backwards: it is not harmonic because it is divergent; it is divergent because it is harmonic. (Lots of divergent series are not at all harmonic.)

Ray Vickson said:
Note: the series cannot be ##\sum_{k=0}^{\infty} 1/k## because the first term would be 1/0. However, starting at ##k = 1## is OK.

Do you mean that one of the series is ##\sum_{k=1}^{\infty} 1/k## and the other is ##\sum_{k=1}^{\infty}(\frac{1}{k}+1)##? That is what you wrote. Did you really mean ##\sum_{k=1}^{\infty} 1/(k+1)## for the second series? If so, use parentheses, like this: 1/(k+1).

Anyway, just write out the first first few terms of both of your series, to see how they are related.

When you say "it's also harmonic series , because the sum is divergent" you have it backwards: it is not harmonic because it is divergent; it is divergent because it is harmonic. (Lots of divergent series are not at all harmonic.)
i mean second one , IMO , it is also harmonic ... , am i right ?

foo9008 said:
i mean second one , IMO , it is also harmonic ... , am i right ?

Have you tried to write out the first few terms of both series to see how they differ? If you do, you can answer your own question.

The infinite sum of 1/(k+1), with k starting from 1, is the same as the infinite sum of 1/k, with k starting from one minus 1. Follow Ray Vickson's advice to see this.

foo9008