Q: What if my index is m and is in the exponent of some "not index" variable?

Then you would still substitute the "m" values into the sum. The base variable would remain unchanged.

Q: Can the index only increase by 1, or is it possible for an index to increase by a larger number?

You can decide that. you give it a starting point and ending point, but you can use terms in the function to make it go faster or slower. You could say the sum from i=0 to n of something with i^2. or you could even have something to the i power. But you will go through the integers with i, yes.So any manipulation of skipping numbers would have to be done in the function. For instance if you only wanted even numbers you would have a 2i in the function.

Q: Additionally, does the "i" simply go up by integer values? What if I wanted a different manner of arithmetically changing "i"?

Some books/teachers will accept writing, for example, 'step 0.5' to indicate that i is going up in increments of 1/2. However, this isn't necessary, since you can just change the expression inside the sum to get the same effect. So if you have the sum from i=1 to 10 of i², and you want to i to step by 1/3 instead of 1, you can change the expression i² to (1+(i-1)/3)², and have i go from i=1 to 28. This will get you the same sequence of numbers as just changing the step.

Question 1

can there be 2 sigma symbols at a time
and what if such condition occurs

Accepted Answer

Yes, there can. Something like Σ³ᵢ₌₁Σ⁴ⱼ₌₁ ij

In this case, we evaluate the innermost (rightmost) sum first. In the end, this will give us a function of i, which we then compute normally.

The inner sum is i+2i+3i+4i, or 10i. So this simplifies to Σ³ᵢ₌₁ 10i, or 10+20+30=60.

Question 2

In the first section (Unpacking Sigma Notation), I've seen the index equal 0. But my calculus teacher says that the index can't be 0, because you can't have the 0th term of a sequence. But all else being equal (the sequence and summation index remaining the same), what would be the difference between a sum with i = 0 and a sum with i = 1?

Thank you.

Thank you.

Accepted Answer

Nothing really. Nothing changes if you shift all the indices down by 1. In fact, you can really start at any index you want because there's no convention that the subscript has to denote which number the term is in the sequence. Generally, people start at index 1 because it happens to be convenient to use the subscripts (and so the indices) to keep track of the number of the terms.

Question 3

How do i derive the formula for summation?

Sum from k to n i = [(n-k+1)(n+k)]/2

Accepted Answer

Another way to derive this formula is to let 
S = Sum from k to n of i, write this sum in two ways, add the equations, and finally divide both sides by 2. We have

S = k + (k+1) + ... + (n-1) + n
S = n + (n-1) + ... + (k+1) + k.

When we add these equations, we get 2S on the left side, and n-k+1 column sums that are each n+k on the right side.

So 2S = (n-k+1)(n+k).  
Dividing both sides by 2 gives S = [(n-k+1)(n+k)]/2.

Question 4

Why is (sigma; n=1 to 3) (n!) -n undefined?

Accepted Answer

I have no idea, it doesn't look like it should be undefined. If we just calculate the sum from 1 to 3, we get a perfectly defined number:
Sum = (1! - 1) + (2! - 2) + (3! - 3)
 = 0 + 0 + 3 = 3
If you set n to infinity though, the series will diverge and there will be no sum. Could you be confusing it with that?

Question 5

What would a function like this look like in Khan Academy Javascript?

Accepted Answer

Basic summation demo with example in article:

```
// Your function
function f(x){ return 2*x - 1; }

// Summation
function summation(lower, upper, f){
    var sum = 0;
    for(var i = lower; i <= upper; i++){
        sum += f(i);
    }
    return sum;
}

// Result gives 9, which is equal to what's expected.
// Arguments: lower bound on x, upper bound on x, f(x)
println(summation(1, 3, f));
```
You can modify the function 'f' in any way you want.

Question 6

What if my index is m and is in the exponent of some "not index" variable?

Accepted Answer

Then you would still substitute the "m" values into the sum. The base variable would remain unchanged.

Question 7

Can the index only increase by 1, or is it possible for an index to increase by a larger number?

Accepted Answer

You can decide that.  you give it a starting point and ending point, but you can use terms in the function to  make it go faster or slower.

You could say the sum from i=0 to n of something with i^2.  or you could even have something to the i power.  But you will go through the integers with i, yes.So any manipulation of skipping numbers would have to be done in the function.  For instance if you only wanted even numbers you would have a 2i in the function.

Question 8

Additionally, does the "i" simply go up by integer values? What if I wanted a different manner of arithmetically changing "i"?

Accepted Answer

Some books/teachers will accept writing, for example, 'step 0.5' to indicate that i is going up in increments of 1/2.
However, this isn't necessary, since you can just change the expression inside the sum to get the same effect.

So if you have the sum from i=1 to 10 of i², and you want to i to step by 1/3 instead of 1, you can change the expression i² to (1+(i-1)/3)², and have i go from i=1 to 28. This will get you the same sequence of numbers as just changing the step.

Course: Integral Calculus > Unit 1

Summation notation

Unpacking the meaning of summation notation

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