Sum to infinity

[Leah5467]

Hello,4b,4c are the questions that i don't know:

answer:


i don't get why -1<log<1. What i am thinking is if infinity means infinite terms and infinite answers,then numbers larger than 1 can still do the same. And what is "sum to infinity"? Does it mean infinite terms? Any examples of "sum to infinity"?

And

Thank you!

 

[pka]

Hello,4b,4c are the questions that i don't know:
View attachment 12226
answer:
i don't get why -1<log<1. What i am thinking is if infinity means infinite terms and infinite answers,then numbers larger than 1 can still do the same. And what is "sum to infinity"? Does it mean infinite terms? Any examples of "sum to infinity"?

First the important fact is: \(\displaystyle \large{0<\log_{10}}(2)<1\). SEE HERE
The sum \(\displaystyle \sum\limits_{k = J}^\infty {A{r^k}} \) is a geometric series that converse if and only if \(\displaystyle \Large{|r|<1} \)
And the sum is \(\displaystyle \sum\limits_{k = J}^\infty {A{r^k}} =\large{ \frac{{A{r^J}}}{{1 - r}}}\)

 

[Leah5467]

Thank you for your detailed answer! I have some things that i don't really understand:Why is 0<log10(2)<1 infinity? And what are A,r and J in Ar? Thank you!

 

[pka]

Thank you for your detailed answer! I have some things that i don't really understand:Why is 0<log10(2)<1 infinity? And what are A,r and J in Ar?

If that is your situation then you need a face-to-face sit down with an instructor.

 

[Leah5467]

Okay thank you!

 

[Harry_the_cat]

Why is 0<log10(2)<1 infinity?

If \(\displaystyle a^x = N \) then \(\displaystyle log_aN=x\)

eg

\(\displaystyle 2^3 =8\) and \(\displaystyle log_{2}8 = 3\)

In other words \(\displaystyle log_{2}8 \) is the power you have to raise 2 to, to get 8 (that is, 3).

So \(\displaystyle log_{10}2\) is the power you have to raise 10 to, to get 2.

You know that \(\displaystyle 10^0 =1\) and \(\displaystyle 10^1 = 10\)

So the power you have to raise 10 to, to get 2 (ie \(\displaystyle log_{10}2\)) must be between 0 and 1.