Melodious Progression
1+1/2+1/3+1/4+1/5+...
If one takes the sum of the first 2 terms, then the first 3, then the first 4, and so on and so forth, at what rate does the partial sum increase?
Does the series converge to 1 finite value, or does it diverge to an infinite value (oscillation also falls under divergence).
Hint: If you don't understand the problem, search up what the harmonic sequence is.
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1 solution
For those of you who do not know Calculus, there is a popular algebraic approach (that is based in the Comparison Test from Calculus) (If you do know Calculus, go to the bottom for a solution using the Integral Test) that you can use to figure out whether the function converges or diverges.
If you look at the series without any fancy notation, like summation, and just the terms, you'll get this:
1, 1/2, 1/3, 1/4, 1/5, 1/6, 1/7, 1/8, 1/9,...
After that, you can find another sequence that has all of it's terms less than or equal to this:
1, 1/2, 1/4, 1/4, 1/8, 1/8, 1/8, 1/8, 1/16,...
This sequence finds the largest power of 2 that is less than or equal to the number that corresponds to it on top.
Now, we can sum the second sequence (You lump all of the common fractions together)
1+1/2+1/2+1/2+1/2+... = infinity
Because 1+1/2+1/3+1/4+... has all terms greater than or equal to the second sequence, the sum of our original sequence must be greater than infinity, or in other words, just infinity.
When a series sums to infinity, you would call it a divergent series.
Without using Calculus, you cannot know for sure how fast the series changes, but you can guess easily by analyzing the functions given. Our series starts off increasing quickly, then slows down. Looking at our options:
Logarithm: Moves very quickly at the beginning, then slows down.
Exponent: Starts slowly, then moves quickly.
Factorial: Starts slowly, the moves quickly.
Linear: Moves at a constant rate throughout.
The choice that best fits our series is Logarithm. Therefore our answer is
Logarithm; Divergent
For those who know Calculus: To check whether this series converges, we can use multiple methods, but the best one for this question would be the integral test.
The function 1/x is a decreasing, positive, and continuous function for all natural numbers (numbers in question). So, now that the prerequisites have been 'approved', we can apply the integral test.
The integral of 1/x is ln(x), and we have to take the limit to infinity of that. If we look at the graph of ln(x), it will increase without bound to infinity as you approach positive infinity on the x axis.
Therefore, our answer is
Logarithm; Divergent
However, this also solves our problem about how fast the series changes. It changes at a logarithmic (since ln(x) is log with a base of e) rate, and is a divergent series.
As much as this seems unintuitive, there are several more mathematical proofs that you can find online that still prove this.