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

[foo9008]

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 ) ?

Homework Equations

The Attempt at a Solution

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

 

[Ray Vickson]

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 ) ?

Homework Equations

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.)

 

[foo9008]

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 ?

 

[Ray Vickson]

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.

 

[member 587159]

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.