Series: Can you explain why I doing a comprehension test here?

[AlexSendler100%]

Can you please explain why I doing a comprehension test here? I can just remove the An sequence (yes, I know it converges) but why its smaller than zero
.

 

[BigBeachBanana]

Can you please explain why I doing a comprehension test here? I can just remove the An sequence (yes, I know it converges) but why its smaller than zero.

Need more context on what [imath]A_n[/imath] is.

 

[AlexSendler100%]

well the only info about it that its posetive and converges

 

[BigBeachBanana]

well the only info about it that its posetive and converges

One way to maximize a fraction is to minimize the denominator.

 

[AlexSendler100%]

yes i know it but the An its not line an N its a sequence so how can it be smaller than zero?

 

[BigBeachBanana]

yes i know it but the An its not line an N its a sequence so how can it be smaller than zero?

Where does it say [imath]A_n[/imath] is smaller than 0? You said it's positive so it must be [imath]A_n >0[/imath].

 

[AlexSendler100%]

like in the image they have An/An + 2 <= An/2 + 0

 

[BigBeachBanana]

They're not saying [imath]A_n[/imath] is less than 0.

They're saying the entire fraction [imath]\dfrac{A_n}{A_n+2}[/imath] is less than [imath]\dfrac{A_n}{0+2}[/imath]

Back to my comment before, one way to maximize a fraction is to minimize the denominator.

Given [imath]A_n[/imath] is positive or [imath]A_n>0[/imath]. Letting [imath]A_n=0[/imath] will minimize the denominator and in effect maximize the entire fraction.

 

[Dr.Peterson]

like in the image they have An/An + 2 <= An/2 + 0

Do you understand that if [imath]a>b[/imath] then [imath]\dfrac{1}{a}<\dfrac{1}{b}|[/imath]? For instance, [imath]3>2[/imath] so [imath]\dfrac{1}{3}<\dfrac{1}{2}[/imath].

 

[stapel]

Can you please explain why I doing a comprehension test here? I can just remove the An sequence (yes, I know it converges) but why its smaller than zero
.View attachment 36464

What is a "comprehension test"?

Note: Smaller denominator means larger fraction (and larger denominator means smaller fraction).

Assuming that all terms [imath]A_n[/imath] are non-negative, then replacing [imath]A_n[/imath] with zero makes for a smaller expression:

[imath]\qquad 0 \leq A_n \rArr 0 + 2 \leq A_n + 2 \rArr \frac{1}{0+2} \geq \frac{1}{A_n + 2}[/imath]

 

[BigBeachBanana]

What is a "comprehension test"?

Probably meant "comparison".