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Do you know any formulas to evaluate sums of series?

State MATHCOUNTS are coming up, and I really need to find a good method for finding sums of series and sequences. I have scoured the internet about this, and I can't find a single method! It would be great if you guys could post an explanation to solve a problem say, like this: \(\sum_{x=1}^{45} 1/2x\). Also, what is the difference between a series and a sequence? Are there formulas for geometric and arithmetic sequences/series? How can you evaluate an infinite converging series/sequence? I'm totally clueless, and any clues would be unbelievably helpful!

Note by Finn Hulse
3 years, 3 months ago

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I have trouble understanding series, too.

However, from my understanding, I have a few ideas in mind (credibility = basically 0):

\(\rightarrow\) I THINK (please correct me if I'm wrong :D) the sum of an infinite geometric series is \[\frac{a}{1-r},\] where \(a\) represents the first term and \(r\) represents the constant ratio...

\(\rightarrow\) I THINK (please correct me if I'm wrong :D) that the summation that you posted above is means that there are 45 terms, and you are asked to find the sum of the series in which the first term is \(\frac{1}{2}\), and all other terms follow the sequence \[\frac{1}{2}, \frac{1}{2\times2}, \frac{1}{2\times3}, \frac{1}{2\times4}, ..., \frac{1}{2\times45},\] though I have no idea how to sum that. (Telescoping?)

Hope this helps, and I hope to learn more about these convoluted series and sums with you! Michael Diao · 3 years, 3 months ago

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@Michael Diao Awesome! That's actually really cool! Isn't it so weird that there seems to be nothing about these kind of formulas, like, anywhere? These kind of problems pop up ALL the time, but I always have to find some wacky way to solve them. Thanks for the infinite geometric progression formula! Finn Hulse · 3 years, 3 months ago

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@Finn Hulse Yeah no problem :D, I have trouble with some MathCounts concepts too. I still have a lot to learn :D

On a side note, good luck on MathCounts States! (I'm going for MD) Michael Diao · 3 years, 3 months ago

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@Michael Diao I'm going for VA! Maybe I'll see you at Nationals! (Probably not if I can't nail this series stuff ;)). Finn Hulse · 3 years, 3 months ago

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@Finn Hulse I probably won't make Nats :/

Mr. Thomas Luo and the Takoma Park people (cough Will Cui cough) are too overpowered ._. Michael Diao · 3 years, 3 months ago

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@Michael Diao Wait! Will Cui! I just saw a bunch of his solutions on AoPS! Right after you said that! That's hilarious! Finn Hulse · 3 years, 3 months ago

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@Michael Diao Haha. Finn Hulse · 3 years, 3 months ago

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@Michael Diao May I ask how you have Level 5 ratings if you can barely recall the geometric series formula? Sam Thompson · 3 years, 3 months ago

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@Sam Thompson Sorry if I offended you in some way, but ratings and levels do not necessarily indicate how good you truthfully are in a subject. For example, the diagnostic problem for me may have been easier than other problems of the same level, or I may have a deeper understanding of some topics in a subject more than others.

If I don't recall the geometric series formula, that simply means that I am incompetent. I may not deserve my level or rating. Michael Diao · 3 years, 3 months ago

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@Sam Thompson It shows that he's awesome at solving problems, not at remembering formulas. Finn Hulse · 3 years, 3 months ago

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I know to solve recurrences.Also have formulae for that. Prasad Nikam · 3 years, 3 months ago

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@Prasad Nikam What are those? What's the formula? Finn Hulse · 3 years, 3 months ago

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Hi Finn!

There isn't much difference between a sequence and a series. Informally speaking, a sequence is a bunch of numbers, and a series is the sum of the individual terms. There are formulae to calculate the sum of terms of an arithmetic progression, a geometric progression, harmonic progression and arithmetico-geometrico progression. I think you should easily be able to find a good website to explain these. Hope this helps a little. Rohan Rao · 3 years, 3 months ago

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@Rohan Rao Could you give me a link? I mean, I'd really just like to know how to sum arithmetic progressions. Finn Hulse · 3 years, 3 months ago

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@Finn Hulse http://www.askiitians.com/iit-jee-progressions-and-series/arithmetic-progression/

I found this quite useful when I learnt it, since it has useful related links on the sidebar too. But if it doesn't serve your purpose, you could always try Wolfram MathWorld or Wikipedia. Rohan Rao · 3 years, 3 months ago

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@Rohan Rao Oh!!! I get it now! Thanks! Finn Hulse · 3 years, 3 months ago

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