How to interpret summation notation

[simple halo]

Hello, I'm finding it tricky to interpret the summation notation. For instance I know we have a lower imit and upper limit. The lower limit is the nth term at which we start or it's the number in the sequence at which we start?

Similarly, when we talk about the upper limit, it's the nth term we stop or we stop at that number when the sequence reaches this value?

From 1 to i, it seems here the sequence is increasing discretely, is this the case for every sequence or we assume here for instance/ in discrete maths that we consider only natural numbers and things increase by a value of 1?

[green latch]

the index increases discretely for every series

and if you have $i=1$ to $n$ you just sum from $1$ to $n$

[noble edge]

chudcel

[green latch]

start at $1$ and end at $n$

[noble edge]

chudcel

[simple halo]

so it's not the nth term but the final value ?

[green latch]

it's both the nth partial sum and the final index value

[fierce python]

In summation notation, the lower and upper limits are values of the index, not values of the sequence

So for example:

Σ from i=1 to n of a_i

That means:

start with i=1,
then i=2,
then i=3,
...
stop when i=n

So the lower limit tells you the first index value, and the upper limit tells you the last index value

[simple halo]

yeah I see

[fierce python]

But, it doesn't mean to stop when the sequence reaches the number n

[simple halo]

when we take in the first value to be 1 in the series, it's just by coincidence ?

[fierce python]

It just means to stop when the index equals n

[simple halo]

yeah so they are just indices

[fierce python]

For example:

Σ from i=1 to 4 of i^2

That means:

i^2 + 2^2 + 3^2 + 4^2

So the index usually increases by 1 each time the notation says otherwise

Standard summation notation, the index runs through integers/natural numbers

[simple halo]

yeahh I see

thanks !

[simple halo]

By the way, anyone knows how did we get the sum involving the fractions and all on the far right hand side pls

[green latch]

$\sum_{k=0}^{n}k=\frac{n(n+1)}{2}$ which is something you can prove by weak induction

[noble edge]

chudcel

[simple halo]

yep but the thing that gives us 1/6n^3 etc, i didn't understand how they did that :c

[green latch]

probably through simplifying the fraction

there is actually a formula for $\sum_{k=1}^{n}k^{2}$

[noble edge]

chudcel

[green latch]

i'll just derive it later

[simple halo]

oh ok

yeah I see, it's all formulae, if you remember the name later on can you pls let me know, I can just have a glance at it later on

.solved