Tweaking the Harmonic

Calculus Level 1

It is known that the harmonic series

$1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} +\cdots$

diverges. What about if we only took every tenth term; does it still diverge?

$\frac{1}{10} + \frac{1}{20} + \frac{1}{30} + \frac{1}{40} + \cdots$

It converges It still diverges

5 solutions

Tom Engelsman
Jan 25, 2017

If the harmonic series is $H$ , then the tweaked series is just $\frac{1}{10} \cdot H$ . Thus, the new series still diverges.

Did Jason Dyer mean the sum of every tenth term of the harmonic series, or the original harmonic series minus every tenth term? No matter, between the two of us, we've got all the bases covered.

Michael Mendrin - 4 years, 4 months ago

It explicitly gives 1/10 + 1/20 + 1/30 + ... in the problem.

Jason Dyer Staff - 4 years, 4 months ago

Like I said, we've got both interpretations covered. I read it quickly and assumed the case that is not so trivial. Consider, the harmonic series diverges, and so does the " $\frac{1}{10}$ " version of it. So, basically $\infty-\infty=?$ . Usually that is divergent too, but not always.

Michael Mendrin - 4 years, 4 months ago

@Michael Mendrin – I suppose the other question would be fine too, if you want to write it? This one is part of my effort to make some level 1-2ish Calculus questions (we're very short on those). If it's indeed easy enough to land at level 1 I met my goal.

Jason Dyer Staff - 4 years, 4 months ago

@Jason Dyer – I have an idea of a problem based on this, which I'll get around to doing it. First, let me think about it.

Michael Mendrin - 4 years, 4 months ago

@Michael Mendrin – I believe Mendrin made a good point about the rigor of the proof. Even though I believe the same that $k \cdot H$ is divergent where $H$ is a harmonic series and $k$ is nonzero real constant.

Interesting point you brought up about $\infty - \infty$ !

Michael Huang - 4 years, 4 months ago

Zach Abueg
Jan 26, 2017

$\displaystyle \frac {1}{10} + \frac {1}{20} + \frac {1}{30} + \frac {1}{40} + ... = \sum_{n=1}^{\infty} \frac{1}{10n}$

Knowing that this and the harmonic series are positive series, let us use the limit comparison test and compare this series to the harmonic series $\displaystyle \sum_{n=1}^{\infty} \frac 1n$ .

$\displaystyle \lim_{x\to\infty} \frac {\frac {1}{10n}}{\frac 1n} = \frac {1}{10}$

$\displaystyle \frac {1}{10} \in \mathbb {R^+}$

Because $\displaystyle \lim_{x\to\infty} \frac {a_n}{b_n} \in \mathbb {R^+}$ , and the harmonic series diverges, we can conclude that $\displaystyle \sum_{n=1}^{\infty} \frac{1}{10n}$ also diverges.

Michael Mendrin
Jan 25, 2017

Consider the series

$\dfrac { 1 }{ 9 } +\dfrac { 1 }{ 19 } +\dfrac { 1 }{ 29 } +\dfrac { 1 }{ 39 } +...>\dfrac { 1 }{ 10 } +\dfrac { 1 }{ 20 } +\dfrac { 1 }{ 30 } +\dfrac { 1 }{ 40 } +...=\dfrac { 1 }{ 10 } \left( 1+\dfrac { 1 }{ 2 } +\dfrac { 1 }{ 3 } +\dfrac { 1 }{ 4 } +... \right)$

which diverges, and that's all we need to prove that the harmonic series minus every tenth term is still divergent.

A Former Brilliant Member
Feb 2, 2017

1/10 + 1/20 + (1/30 + 1/40) + (1/50 + 1/60 + 1/70 + 1/80) + (1/90 + 1/100 + 1/110 + 1/120 + 1/130 + 1/140 + 1/150 + 1/160) + ...

> 1/10 + 1/20 + (1/40 + 1/40) + (1/80 + 1/80 + 1/80 + 1/80) + (1/160 + 1/160 + 1/160 + 1/160 + 1/160 + 1/160 + 1/160 + 1/160) + ...

= 1/10 + 1/20 + 2(1/40) + 4(1/80) + 8(1/160) + ...

= 1/10 + 1/20 +1/20 + 1/20 + 1/20 + ... = infinity

Nitish Kumar
Jan 28, 2017

Take 1/10 common.so the inside part is same as the harmonic series given.hence,it diverges.

Tweaking the Harmonic

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