What is the sum of value of S?

john458776

New member
Joined
May 15, 2021
Messages
25
20210610_142633.jpg
This is how solved it. But I'm not sure if 2.2 is correct.
2.1
\(\displaystyle S\)1 \(\displaystyle = 3\)
\(\displaystyle S\)2 \(\displaystyle = 9/2\)
\(\displaystyle S\)3 \(\displaystyle = 43/9\)
\(\displaystyle S\)4 \(\displaystyle = 39/8\)
\(\displaystyle S\)5 \(\displaystyle = 123/25\)


2.2
\(\displaystyle S\)infinity\(\displaystyle = 5-2/\)(infinity)2 \(\displaystyle = 5\)
 
Last edited:

lev888

Senior Member
Joined
Jan 16, 2018
Messages
2,153
View attachment 27723
This is how solved it. But I'm not sure if 2.2 is correct.
2.1
\(\displaystyle S\)1 \(\displaystyle = 3\)
\(\displaystyle S\)2 \(\displaystyle = 9/2\)
\(\displaystyle S\)3 \(\displaystyle = 43/9\)
\(\displaystyle S\)4 \(\displaystyle = 39/8\)
\(\displaystyle S\)5 \(\displaystyle = 123/25\)


2.2
\(\displaystyle S\)infinity\(\displaystyle = 5-2/\)(infinity)2 \(\displaystyle = 5\)
What's the relationship between the sum of the series and the partial sums sequence?
 

john458776

New member
Joined
May 15, 2021
Messages
25
The partial sums of a series form a new sequence, which is denoted as {s1, s2, s3, s4,...}.
What's the relationship between the sum of the series and the partial sums sequence?
 

pka

Elite Member
Joined
Jan 29, 2005
Messages
11,003
View attachment 27723
This is how solved it. But I'm not sure if 2.2 is correct.
2.1
\(\displaystyle S\)1 \(\displaystyle = 3\)
\(\displaystyle S\)2 \(\displaystyle = 9/2\)
\(\displaystyle S\)3 \(\displaystyle = 43/9\)
\(\displaystyle S\)4 \(\displaystyle = 39/8\)
\(\displaystyle S\)5 \(\displaystyle = 123/25\)

2.2\(\displaystyle S\)infinity\(\displaystyle = 5-2/\)(infinity)2 \(\displaystyle = 5\)
There is no such number as \(\frac{1}{\infty^2}\)
However, it is true that \(\mathop {\lim }\limits_{N \to \infty } \frac{2}{{{N^2}}} = 0\)
 

john458776

New member
Joined
May 15, 2021
Messages
25
View attachment 27723
This is how solved it. But I'm not sure if 2.2 is correct.
2.1
\(\displaystyle S\)1 \(\displaystyle = 3\)
\(\displaystyle S\)2 \(\displaystyle = 9/2\)
\(\displaystyle S\)3 \(\displaystyle = 43/9\)
\(\displaystyle S\)4 \(\displaystyle = 39/8\)
\(\displaystyle S\)5 \(\displaystyle = 123/25\)


2.2
\(\displaystyle S\)infinity\(\displaystyle = 5-2/\)(infinity)2 \(\displaystyle = 5\)
I think
2.1 Should be in this form {3, 9/2, 43/9, 39/8, 123/25}
 

lev888

Senior Member
Joined
Jan 16, 2018
Messages
2,153

john458776

New member
Joined
May 15, 2021
Messages
25
There is no such number as \(\frac{1}{\infty^2}\)
However, it is true that \(\mathop {\lim }\limits_{N \to \infty } \frac{2}{{{N^2}}} = 0\)
Yes, I think I made a mistake there by not including the limit.
 

john458776

New member
Joined
May 15, 2021
Messages
25
What's the relationship between the sum of the series and the partial sums sequence?
I thin when we add the sum of the series, we obtain the partial sum sequences
For example a series : 1+2+3+...
When we add the S2=1+2=3, we obtain the partial sum.
According to my understanding.
 

lev888

Senior Member
Joined
Jan 16, 2018
Messages
2,153
I thin when we add the sum of the series, we obtain the partial sum sequences
For example a series : 1+2+3+...
When we add the S2=1+2=3, we obtain the partial sum.
According to my understanding.
When do we know that an infinite series has a finite sum? And how do we find it?
 

JeffM

Elite Member
Joined
Sep 14, 2012
Messages
6,430
You missed the point of question 2.1.

\(\displaystyle n \in Z^+ \implies S_n < S_{n+1} < 5\)

may be intuitively obvious, but it is vividly demonstrated by stating 2.1 in a different form

\(\displaystyle S_1 = 5 - \dfrac{2}{1} = 4 - 1 = 3 < 5\\
S_2 = 5 - \dfrac{2}{4} = 4 + \dfrac{2}{4} < 5\\
S_3 = 5 - \dfrac{2}{9} = 4 + \dfrac{7}{9} < 5\\
S_4 = 5 - \dfrac{2}{64} = 4 + \dfrac{62}{64} < 5\\
S_5 = 5 - \dfrac{2}{125} = 4 + \dfrac{123}{125} < 5.\)
Moreover that makes it easy to see that the series is quickly converging toward 5.

Consider S100.

\(\displaystyle S_{100} = 5 - \dfrac{2}{10000} = 5 - 0.0002 = 4.9998 \approx 5.\)

Now that is not a proof, but it should motivate a proof that the series converges to 5.
 

john458776

New member
Joined
May 15, 2021
Messages
25
When do we know that an infinite series has a finite sum? And how do we find it?
If the sequence of partial sums does not have a real limit, we say the series does not have a sum.
 

john458776

New member
Joined
May 15, 2021
Messages
25
View attachment 27723
This is how solved it. But I'm not sure if 2.2 is correct.
2.1
\(\displaystyle S\)1 \(\displaystyle = 3\)
\(\displaystyle S\)2 \(\displaystyle = 9/2\)
\(\displaystyle S\)3 \(\displaystyle = 43/9\)
\(\displaystyle S\)4 \(\displaystyle = 39/8\)
\(\displaystyle S\)5 \(\displaystyle = 123/25\)


2.2
\(\displaystyle S\)infinity\(\displaystyle = 5-2/\)(infinity)2 \(\displaystyle = 5\)
Since it tends to infinity. So the series
does not have a sum Okay.
 

Cubist

Full Member
Joined
Oct 29, 2019
Messages
928
I'm a bit confused. I kind of agree with the answer "5" in the first post, subject to the important correction that @pka made in post #4.
 

JeffM

Elite Member
Joined
Sep 14, 2012
Messages
6,430
Since it tends to infinity. So the series
does not have a sum Okay.
It does not tend to infinity. It never exceeds 5. The only question is whether it converges to a finite number and if so what that number is.
 

Cubist

Full Member
Joined
Oct 29, 2019
Messages
928
The wording of the question 2.2 seems pretty strange to me, "What is the sum of value of S?".

I interpret it to be "what is the value of S". (Obviously S is itself a sum).
 

lev888

Senior Member
Joined
Jan 16, 2018
Messages
2,153
If the sequence of partial sums does not have a real limit, we say the series does not have a sum.
I asked when the series does have a sum (and how to find it). Isn't this the question you need to answer to solve the problem?
 

john458776

New member
Joined
May 15, 2021
Messages
25
The wording of the question 2.2 seems pretty strange to me, "What is the sum of value of S?".

I interpret it to be "what is the value of S". (Obviously S is itself a sum).
Value of S is 5
And the series Converges since the limit is finite.
 
Top