Can anyone explain this formula to me?

KFS

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What does ai, ai-1, an, and a0 mean? I studied this subject without getting too deep in telescoping sums because I don't understand this. Can anyone explain to me? Any help will be welcome, thank you.

sum.jpg
 
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write out first -say 5 terms:

= (a1 - a0) + (a2 - a1) + (a3 - a2) + (a4 - a3) .....

Do you see a pattern?
 
Yes but how do you apply that when it comes to a given function?
 
Yes but how do you apply that when it comes to a given function?
\(\displaystyle \begin{align*}\sum\limits_{k = 1}^{18} {\frac{1}{{{x^2} + 3x + 2}}} &= \sum\limits_{k = 1}^{18} {\frac{1}{{x + 2}} - \frac{1}{{x + 1}}} \\&=\left(\dfrac{1}{3}-\dfrac{1}{2}\right)+\left(\dfrac{1}{4}-\dfrac{1}{3}\right)+\left(\dfrac{1}{5}-\dfrac{1}{4}\right)+\cdots +\left(\dfrac{1}{20}-\dfrac{1}{19}\right)\\&=\left(\dfrac{1}{20}-\dfrac{1}{2}\right) \end{align*}\)
 
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What does ai, ai-1, an, and a0 mean? I studied this subject without getting too deep in telescoping sums because I don't understand this. Can anyone explain to me? Any help will be welcome, thank you.
a0 is the 1st term. Note immediately that the subscript is 1 less than term number.
So ai is the (ith + 1st) term, an is the (nth + 1st) term.

Note: Some books start with a1, not a0. So a1is the 1st term and anis the nth term.
You need to see how your book defines things.
 
\(\displaystyle \begin{align*}\sum\limits_{k = 1}^{18} {\frac{1}{{{x^2} + 3x + 2}}} &= \sum\limits_{k = 1}^{18} {\frac{1}{{x + 2}} - \frac{1}{{x + 1}}} \\&=\left(\dfrac{1}{3}-\dfrac{1}{2}\right)+\left(\dfrac{1}{4}-\dfrac{1}{3}\right)+\left(\dfrac{1}{5}-\dfrac{1}{4}\right)+\cdots +\left(\dfrac{1}{20}-\dfrac{1}{19}\right)\\&=\left(\dfrac{1}{20}-\dfrac{1}{2}\right) \end{align*}\)
Professor, I check three times to make sure that you 1st line was in fact wrong. It seems that you are off by a negative sign.
 
Professor, I check three times to make sure that you 1st line was in fact wrong. It seems that you are off by a negative sign.
Professor, I check three times to make sure that you 1st line was in fact wrong. It seems that you are off by a negative sign.
You are correct. I caught that also. But the new damn time limit prevented me from correcting it.
It should be
\(\displaystyle \begin{align*}\sum\limits_{x = 1}^{18} {\left( {\frac{1}{{{x^2} + 3x + 2}}} \right)} &= \sum\limits_{x = 1}^{18} {\left( {\frac{1}{{x + 1}} - \frac{1}{{x + 2}}} \right)}\\&=\left( {\frac{1}{2} - \frac{1}{3}} \right) + \left( {\frac{1}{3} - \frac{1}{4}} \right) + \cdots + \left( {\frac{1}{{19}} - \frac{1}{{20}}} \right) \\&= \left( {\frac{1}{2} - \frac{1}{{20}}} \right)\end{align*}\)
 
You are correct. I caught that also. But the new damn time limit prevented me from correcting it.
It should be
\(\displaystyle \begin{align*}\sum\limits_{x = 1}^{18} {\left( {\frac{1}{{{x^2} + 3x + 2}}} \right)} &= \sum\limits_{x = 1}^{18} {\left( {\frac{1}{{x + 1}} - \frac{1}{{x + 2}}} \right)}\\&=\left( {\frac{1}{2} - \frac{1}{3}} \right) + \left( {\frac{1}{3} - \frac{1}{4}} \right) + \cdots + \left( {\frac{1}{{19}} - \frac{1}{{20}}} \right) \\&= \left( {\frac{1}{2} - \frac{1}{{20}}} \right)\end{align*}\)
Now ... now ...
 
Now ... now ...
pka is a retired research mathematician! That says it all (at least for me). If he wants to say damn, I give him permission. pka deserves whatever entitlements he wants in order to volunteer on this forum.

Subhotosh, harass Denis instead. And remove his posts again
 
So I start from the beginning, I make sure cancelation, and those that remain (the first and the last one) is the solution. Is that correct?
 
So I start from the beginning, I make sure cancelation, and those that remain (the first and the last one) is the solution. Is that correct?
IF it is a telescoping series
 
pka is a retired research mathematician! That says it all (at least for me). If he wants to say damn, I give him permission. pka deserves whatever entitlements he wants in order to volunteer on this forum.
nd remove his posts agai
No, he doesn't deserve "whatever entitlements" than anyone else. He's just another username on the board. Don't give attributions to
usernames on the board. In fact, the more of an elitist/arrogant attitude that some username displays here, the more that username
needs to be taken down notches. That goes with any helper.

What some username actually accomplishes here by way of correctness toward the problems of assisting and their attitude/tone
are a couple of the major things that matter. People's accomplishments outside (e.g. formal education) of here are irrelevant.
Don't throw outside accomplishments up in front of anyone's faces or allow them/expect them to receive special favors or considerations
because of them.
 
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lookagain, pka (and even yourself) is a great addition to this forum. pka can solve every problem posted here and do it correctly. That deserves pka some entitlement. I am just saying we should not say a word if pka uses a word that may be inappropriate. After all, all I am saying is to allow him to say a word, not very much in my opinion.
 
lookagain, pka (and even yourself) is a great addition to this forum. pka can solve every problem posted here and do it correctly. That deserves pka some entitlement. I am just saying we should not say a word if pka uses a word that may be inappropriate. After all, all I am saying is to allow him to say a word, not very much in my opinion.
You know what I say to that? ?

-Dan
 
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