Diff/ Sum Question Pls Help (1 Viewer)

[eternallyboreduser]

Why do we the let the number of top of sigma be k? Why not infinity? Just wondering, thanks in advance

 

[eternallyboreduser]

@WeiWeiMan @liamkk112 @carrotsss Work your magic pls

 

[liamkk112]

Why do we the let the number of top of sigma be k? Why not infinity? Just wondering, thanks in advance View attachment 42234View attachment 42235View attachment 42237

well if its an infinite sum, then u should take the limit as k goes to infinity

their working out is kinda janked though but prob previous parts make sense

 

[eternallyboreduser]

well if its an infinite sum, then u should take the limit as k goes to infinity

their working out is kinda janked though but prob previous parts make sense

Wdym? My teacher said to create a telescoping sum, but im wondering why they put k at the top

 

[eternallyboreduser]

Bump

 

[kendricklamarlover101]

k is just an arbitrary value u can use any letter as long as u take the limit to infinity

 

[eternallyboreduser]

k is just an arbitrary value u can use any letter as long as u take the limit to infinity

When you say taking the limit to infinity youre talking about subbing in all the x values right for example like 1+2+3+....+...+n
So like this part

 

[eternallyboreduser]

k is just an arbitrary value u can use any letter as long as u take the limit to infinity

But why dont we let it be infinity then? Is it cos it would be hard to form a telescoping sum that way?

 

[kendricklamarlover101]

But why dont we let it be infinity then? Is it cos it would be hard to form a telescoping sum that way?

infinity isnt a number most of the time when u see infinities in math ur actually just taking a limit to infinity. the infinity commonly used in math is just another way of saying what happens as this number gets really big.

u can still make a telescoping sum if the symbol used is infinity it doesnt really matter

 

[eternallyboreduser]

infinity isnt a number most of the time when u see infinities in math ur actually just taking a limit to infinity. the infinity commonly used in math is just another way of saying what happens as this number gets really big.

u can still make a telescoping sum if the symbol used is infinity it doesnt really matter

But then wouldnt u sub in infinity which dorsnt rlly make sense and i dont think it would gey u the right answer either

 

[eternallyboreduser]

When you say taking the limit to infinity youre talking about subbing in all the x values right for example like 1+2+3+....+...+n
So like this part

? @kendricklamarlover101 Is that what you meant?

 

[eternallyboreduser]

? @kendricklamarlover101 Is that what you meant?

?

 

[kendricklamarlover101]

But then wouldnt u sub in infinity which dorsnt rlly make sense and i dont think it would gey u the right answer either

if u do take the limit u get that the result would equal 1 + ln(2) + ln(2) > 1+ln2

? @kendricklamarlover101 Is that what you meant?

u can take the limit from the start just by writing

and if u want to expand the sum out to see the telescoping series the result would look like


also im pretty sure the solution has an error as the first 2 sums should just cancel out to ln(k).

 

[eternallyboreduser]

if u do take the limit u get that the result would equal 1 + ln(2) + ln(2) > 1+ln2


u can take the limit from the start just by writing

and if u want to expand the sum out to see the telescoping series the result would look like


also im pretty sure the solution has an error as the first 2 sums should just cancel out to ln(k).

How exactly would you solve that eqn tho? Is it the same way as the written sol?

 

[kendricklamarlover101]

How exactly would you solve that eqn tho? Is it the same way as the written sol?

yup basically just adding that limit notation to show that ur actually taking the infinite sum

 

[eternallyboreduser]

yup basically just adding that limit notation to show that ur actually taking the infinite sum

Oh, so in the solution where they write the sum and let the top be k theyre essentially saying its infinity but they didnt writr the limit

 

[kendricklamarlover101]

Oh, so in the solution where they write the sum and let the top be k theyre essentially saying its infinity but they didnt writr the limit

yup

 

[kendricklamarlover101]

lol these emojis are so silly like has anybody ever used

 

[eternallyboreduser]

yup

Oh okay, thanks alot for the clarification

 

[Luukas.2]

The first line on the second page, where the is introduced, is flawed. It states that

​

which may or may not be true, depending on the value of :

​

The proper way to introduce the is to also introduce a limit, as the LHS is also an infinite sum:

​

or, to take a finite sum on both sides:

​

and then take limits as at the end.

The solution shown from the answers is sloppily written in how it introduces the limit and provides an example that should not be followed. Whether it would be penalised would depend on the marker. I would probably tolerate it, but grizzle, if the line stated something like "for large "... but as it stands, I have shown two values of for which the result is provably false, and I am sure there are more if I include more terms on the LHS.

Incidentally, the expression on the LHS was shown by Euler to be:

​

which established the required result as

​