Another question needed btw Thanks Enigma for the reply (1 Viewer)

[ssukarieh1]

Hello, limiting sums question from 3 Unit Cambridge, last question in the exercise.
17. Find the condition for each GP to have a limiting sum, then find that limiting sum:
a) 1+(x^2-1)+(x^2-1)^2+...

g) 1+2x/1+x^2+ (2x)^2/(1+x^2)^2+...

I understand that -1< r< 1, however, when I substitute r, I keep getting the wrong answer. The answer to
a) is -(2^1/2) < x < (2^1/2) if that helps.

Thank you , much appreciated!!

 

[ssukarieh1]

Yes I know for the second part of the question, but that doesn't apply to the first part, which asks for the condition for the GP to have a limiting sum

 

[enigma_1]

no worries!

open this View attachment 31298

bos is not letting me write the inequalities properly so I had to attach it lel

then limitins sum is = a/(1-r)

= 1/(1-(x^2 -1))
= 1/(2-x^2)

 

[ssukarieh1]

Thanks Enigma!!! But what about question g) ??

 

[braintic]

no worries!

open this View attachment 31298

bos is not letting me write the inequalities properly so I had to attach it lel

When you solve 0 < x^2 <2, you have to exclude x=0 from your solution.

 

[enigma_1]

Thanks Enigma!!! But what about question g) ??

no worries )) for 17g first you find the limiting sum which is (x^2 +1)/(x-1)^2 and then the denominator cannot be equivalent to zero so the condition x=/= 1 and x=/= -1 should be stated for the GP to have a limiting sum

 

[enigma_1]

When you solve 0 < x^2 <2, you have to exclude x=0 from your solution.

this

sorry didn't realise it, it's coz if x=0 then r would be equal to r=-1 which cannot occur for a limiting sum to occur

 

[ssukarieh1]

1+2x/(1+x^2)+ (2x)^2/(1+x^2)^2+... Like that?

 

[ssukarieh1]

Would you consider this somewhat of a difficult question for a 2 Unit student?

 

[braintic]

1+2x/(1+x^2)+ (2x)^2/(1+x^2)^2+... Like that?

-1 < 2x/(1+x^2) < 1

-1-x^2 < 2x < 1+x^2

LHS:
x^2 + 2x + 1 >0
(x+1)^2 > 0
x not equal to -1

RHS:
x^2 -2x +1 > 0
(x-1)^2 >0
x is not equal to 1


S = 1 / [1 - 2x/(1+x^2) ]

= (1+x^2) / (1+x^2 - 2x)

= (1+x^2) / (1-x)^2

 

[braintic]

no worries )) for 17g first you find the limiting sum which is (x^2 +1)/(x-1)^2 and then the denominator cannot be equivalent to zero so the condition x=/= 1 and x=/= -1 should be stated for the GP to have a limiting sum

The restriction x not equal to -1 doesn't come from the denominator.

 

[enigma_1]

The restriction x not equal to -1 doesn't come from the denominator.

My bad

 

[seventhroot]

you should probably ask square

 

[Queenroot]

you should probably ask square

use the formula S = a/(1-r)