# What is the sum of value of S?

#### john458776

##### New member

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
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
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
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
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
The partial sums of a series form a new sequence, which is denoted as {s1, s2, s3, s4,...}.

#### john458776

##### New member
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
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
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
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
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
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
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
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
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
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
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.