Does the sum of ln(k/(k+1)) converge or diverge as n approaches infinity?

In summary, the conversation discusses the convergence or divergence of the series \sum_{k=1}^{n} ln(\frac{k}{k+1}), with one person stating that it must converge to zero because all terms after ln(1) cancel out, while the other person argues that the series is actually divergent because the last term does not cancel out. The conversation also brings up the question of whether there is always a term bigger than the last, and the expression for the series is given as \sum_{k=1}^{n} ln(\frac{k}{k+1}). The limit of this expression as n tends to infinity is not explicitly stated.
  • #1
cp255
54
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So I was trying to see if [itex]\Sigma[/itex]ln([itex]\frac{n}{n+1}[/itex]) diverges or converges. To see this I started writing out [ln(1) - ln(2)] + [ln(2) - ln(3)] + [ln(4) - ln(5)] ...

I noticed that after ln(1) everything must cancel out so I reasoned that the series must converge on ln(1) which equals ZERO. However, Wolfram Alpha says the series is divergent. I tried looking it up how to do this problem and I read that the last term also does not cancel out which is why the series diverges. However, shouldn't there always be a term bigger than the last so everything must cancel.
 
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  • #2
Write an expression for [itex] \sum_{k=1}^{n} ln(\frac{k}{k+1}) [/itex]. What is the limit of this expression as n tends to infinity?
 

1. What is an infinite sum of logs?

An infinite sum of logs is a mathematical expression that involves taking the logarithm of a number and then adding it to the logarithm of another number, and so on, an infinite number of times. It is written in the form of Σlog(n), where n represents the numbers being added.

2. How do you solve an infinite sum of logs?

To solve an infinite sum of logs, you can use the formula for the sum of an infinite geometric series. This formula is a/(1-r), where a is the first term and r is the common ratio. In the case of an infinite sum of logs, the first term is log(a) and the common ratio is log(b/a). Plug these values into the formula and simplify to find the sum.

3. What is the convergence of an infinite sum of logs?

The convergence of an infinite sum of logs depends on the values of the numbers being added. If the common ratio (log(b/a)) is less than 1, then the sum will converge. If the common ratio is equal to 1 or greater than 1, the sum will diverge.

4. Can an infinite sum of logs be negative?

Yes, it is possible for an infinite sum of logs to be negative. This can happen if the common ratio is between -1 and 0, as this will result in the sum being negative. For example, the infinite sum of logs for log(2) would be -1, as the common ratio (log(2)/log(1)) is -1.

5. What are some real-life applications of infinite sums of logs?

Infinite sums of logs can be used in various fields such as finance, physics, and statistics. In finance, they can be used to calculate compound interest. In physics, they can be used to model exponential decay or growth. In statistics, they can be used to calculate probabilities in a geometric distribution.

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