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Explanation on Notation of Summations And Products

I’ve been wanting to make this post for a long time, since it is severely limiting what I can post on this blog.

These symbols (∑ and ∏) might look very daunting at first but I assure you, even a toddler can understand what it means.

“Hey Fredrick, wanna try a new game?”

 

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Game PosterI spent far too long on those posters. Thanks to Echonox and Cotchy for the design. Posters done using Scrub Microsoft PowerPoint 2012.

Let’s first start with the summation: ∑ | Sigma notation

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The above symbol basically means “To sum” or “Add everything together”. You guys know what is addition right? 1+1=3 ( ͡° ͜ʖ ͡°)? No? Ok nvm.

But sum what with what? Well, let me introduce the dummy variable:

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The dummy variable is basically an instruction of how to sum. For the example on the left, “n” is the dummy variable.

“a” is the starting point

“b” is the end point

and “f(n)” is the function you want to sum over.

What does this all mean?

Well, basically:

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That’s it! Notice how “a” is the starting point and “b” is the end point?

Let’s take a look at some examples:

 

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For the third example, note that “x” is the dummy variable and not y.


Now, what happens if you are walking along a beautiful path on a bright sunny day when suddenly a burly man pulls out a butterfly knife with a pearl-embedded, elephant tusk handle and put it against your throat. Adrenaline rushes through your veins as your limbs go numb with fear while your eyes watch the empty space ahead, futile in making sense of your awaiting doom. With a deep croak, your adversary spat

“We meet again. Now let’s talk.”

He pulls out a slip of paper. The paper cried as it flapped against the wind.

“Take a look. Take a close look at this. Explain.”

The slip of paper makes it’s way to inches from your face. Close up, you piece together the various symbols on the paper and break out in cold sweat. On the paper, largely printed, you saw:

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Now what the hell is that “infinity” symbol doing there? Julian your master failed to cover this in his lectures, and you know you are doomed, for you don’t know what it means.


You don’t want to end up in the situation above do you? So listen up!

The infinity symbol up there means that you sum to infinity. You don’t stop summing, you just add and add and add and add…….. You are adding an infinite number of terms.

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You don’t stop!

You might be asking now, wouldn’t the sum diverge to infinity? In other words:

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Well the answer is: Yes, most of the time. MOST of the time.

You see, for some sums, it doesn’t approach infinity. For instance, you can proof that

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Try it on your calculator! Or… research on Infinite Geometric Progressions… or not.

When it comes to infinite sums, there can be some quite unexpected results. For instance:

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And yes, the π is indeed the one we are familiar with ( π=3.1415…)

To learn more about it, you can research on the Basel Problem, or the Riemann Zeta Function.


There are other types of summations, such as cyclic sum, symmetric sum and sum-over-n-that-satisfies-some-conditions (I’m don’t know the actual name for this type of sum, hopefully somebody can help fill me up on that). Anyway, I’m not going to go through those sums cause I’m lazy. Let’s move on to ∏ (product):

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∏ is essentially the same as ∑, just that instead of summing, you multiply!

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Now let’s take a look at random examples:

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For the second example, note that “n” is the dummy variable and not “x”.

Similar to infinite sums, there are also infinite products, where you multiply FOREVER! HAHAHA!


Now, what happens if you are walking along a beautiful path on a bright sunny day when suddenly a burly man pulls out a butterfly knife with a pearl-embedded, elephant tusk handle and put it against your throat. Adrenaline rushes through your veins as your limbs go numb with fear while your eyes watch the empty space ahead, futile in making sense of your awaiting doom. With a deep croak, your adversary spat

“We meet again. Now let’s talk.”

He pulls out a slip of paper. The paper cried as it flapped against the wind.

“Take a look. Take a close look at this. Tell me. Tell me the exact value.”

The slip of paper makes it’s way to inches from your face. Close up, you piece together the various symbols on the paper and break out in cold sweat. On the paper, largely printed, you saw:

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You faint like a Victorian lady, for you do not know.


Well, for this case, I’ll probably let you die, because I’m too lazy to give you a solution to the above problem. (Also because I’m not sure (Also because I don’t know)). However, I can give you the answer though:

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Basically, my point is, just like an infinite sum, an infinite product can also yield interesting results.


Challenge: Using the above result regarding the infinite product, proof that:

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If the LHS and the RHS exists. Hence, proof that:

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First part of the challenge question can be solved with Sec 4 syllabus while the second part involves A-level syllabus. However, the second part is easier as compared to the first.


I hope that now you know what ∏ and ∑ means! Just don’t go around doing stuff like this:

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Yes Mr.Philosopher, I see you just considered the first term

Or this:

 

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Same guy as above. Yep, I agree 1=1

Thank you for your kind attention

 

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