What is the sum of value of S?

john458776

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This is how solved it. But I'm not sure if 2.2 is correct.
2.1
[MATH]S[/MATH]1 [MATH]= 3[/MATH] [MATH]S[/MATH]2 [MATH]= 9/2[/MATH] [MATH]S[/MATH]3 [MATH]= 43/9[/MATH] [MATH]S[/MATH]4 [MATH]= 39/8[/MATH] [MATH]S[/MATH]5 [MATH]= 123/25[/MATH]

2.2
[MATH]S[/MATH]infinity[MATH] = 5-2/[/MATH](infinity)2 [MATH] = 5[/MATH]
 
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View attachment 27723
This is how solved it. But I'm not sure if 2.2 is correct.
2.1
[MATH]S[/MATH]1 [MATH]= 3[/MATH][MATH]S[/MATH]2 [MATH]= 9/2[/MATH][MATH]S[/MATH]3 [MATH]= 43/9[/MATH][MATH]S[/MATH]4 [MATH]= 39/8[/MATH][MATH]S[/MATH]5 [MATH]= 123/25[/MATH]

2.2
[MATH]S[/MATH]infinity[MATH] = 5-2/[/MATH](infinity)2 [MATH] = 5[/MATH]
What's the relationship between the sum of the series and the partial sums sequence?
 
View attachment 27723
This is how solved it. But I'm not sure if 2.2 is correct.
2.1
[MATH]S[/MATH]1 [MATH]= 3[/MATH][MATH]S[/MATH]2 [MATH]= 9/2[/MATH][MATH]S[/MATH]3 [MATH]= 43/9[/MATH][MATH]S[/MATH]4 [MATH]= 39/8[/MATH][MATH]S[/MATH]5 [MATH]= 123/25[/MATH]
2.2[MATH]S[/MATH]infinity[MATH] = 5-2/[/MATH](infinity)2 [MATH] = 5[/MATH]
There is no such number as \(\frac{1}{\infty^2}\)
However, it is true that \(\mathop {\lim }\limits_{N \to \infty } \frac{2}{{{N^2}}} = 0\)
 
View attachment 27723
This is how solved it. But I'm not sure if 2.2 is correct.
2.1
[MATH]S[/MATH]1 [MATH]= 3[/MATH] [MATH]S[/MATH]2 [MATH]= 9/2[/MATH] [MATH]S[/MATH]3 [MATH]= 43/9[/MATH] [MATH]S[/MATH]4 [MATH]= 39/8[/MATH] [MATH]S[/MATH]5 [MATH]= 123/25[/MATH]

2.2
[MATH]S[/MATH]infinity[MATH] = 5-2/[/MATH](infinity)2 [MATH] = 5[/MATH]
I think
2.1 Should be in this form {3, 9/2, 43/9, 39/8, 123/25}
 
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.
 
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.
 
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?
 
You missed the point of question 2.1.

[MATH]n \in Z^+ \implies S_n < S_{n+1} < 5[/MATH]
may be intuitively obvious, but it is vividly demonstrated by stating 2.1 in a different form

[MATH]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.[/MATH]Moreover that makes it easy to see that the series is quickly converging toward 5.

Consider S100.

[MATH]S_{100} = 5 - \dfrac{2}{10000} = 5 - 0.0002 = 4.9998 \approx 5.[/MATH]
Now that is not a proof, but it should motivate a proof that the series converges to 5.
 
View attachment 27723
This is how solved it. But I'm not sure if 2.2 is correct.
2.1
[MATH]S[/MATH]1 [MATH]= 3[/MATH] [MATH]S[/MATH]2 [MATH]= 9/2[/MATH] [MATH]S[/MATH]3 [MATH]= 43/9[/MATH] [MATH]S[/MATH]4 [MATH]= 39/8[/MATH] [MATH]S[/MATH]5 [MATH]= 123/25[/MATH]

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