Let the positive series ∑ UN converge and the sequence {VN} be bounded. It is proved that the series ∑ UN} is absolutely convergent

Use the comparative discriminant method to prove. The economic mathematics team will help you to answer. Please evaluate in time

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1. Let the positive series ∑ UN diverge and Sn be the partial sum sequence of UN. It is proved that the series ∑ UN / Sn ^ 2 converges
2. If the series ∑ UN converges to s, the series ∑ [UN + UN + 1] converges to s {n from 1 to infinity}
3. If the series UN converges to s, then the series (UN + UN + 1) converges to s
4. A series ∑ UN converges. How can we prove that its odd term ∑ u2n-1 also converges?
5. It is proved that if the series ∑ UN satisfies (1) Limun = 0, (2) ∑ (u2n-1 + u2n) convergence, then ∑ UN converges
6. If ∑ UN converges, does ∑ u2n converge? Conversely, does ∑ u2n converge and ∑ UN converge?
7. Why is the current I equal and the voltage u always = U1 + U2 +... UN when the circuit is in series
8. Given U1 = 1, U2 = 2, find UN = 2U (n-2) + U (n-1) + 1 Solve how to deduce the general term formula
9. Given the partial sum of series Sn = 2n / N + 1, find U1, U2, UN
10. Let U1 = 1, U2 = 1, UN + 1 = 2un + 3un-1 (n = 2,3,...) ) bn=Un/Un+1(n=2,3,…… ）Prove the existence of limbn, and discuss the convergence and divergence of series 1 / UN Here is my own solution The characteristic equation is R ^ 2-2r-3 = 0, R1 = - 1, R2 = 3, the general solution U (n) = C1 * (- 1) ^ n + C2 * 3 ^ n, UN = 3 ^ n-3 * (- 1) ^ n] / 6, then BN = 1 / 3, and then find the reciprocal of BN to get the divergence But I don't think it's right. Eigenvalue? Will solve, the original difference equation, I did not want to be detailed
11. Let the series ∑ (n = 1) UN converge and ∑ UN = u, then the series ∑ (UN + U (n + 1)) =?
12. How many odd divisors are there from 360 to 630?
13. How much is (one third) + two fifths? It's one fifteenth
14. Given AB = 1, a is not equal to 1, find 1 + 1 / A + 1 + 1 / b
15. Let the sum of the first n terms of the sequence {an} be Sn, and 2An = Sn + 2n + 1 (n ∈ n *) (I) find A1, A2, A3; (II) prove that the sequence {an + 2} is an equal ratio sequence; (III) find the sum of the first n terms of the sequence {n · an} and TN
16. In the calculation of the value of (x + y) (x-2y) - my (nx-y) (m.n are all constants) Calculate the value of (x + y) (x-2y) - my (nx-y) (m.n are constants). When substituting the value of X and Y into the calculation, careless Xiaoming and Xiaoliang misjudged the value of Y, but the calculation results were all equal to 25. Careful Xiaomin substituted the correct value of X and y into the calculation, and the result was exactly 25. In order to find out, she arbitrarily took 2009 as the value of Y, and you say it's strange, the result was still 25 (1) According to the above situation, try to explore the mystery; (2) Can you determine the values of M, N and X?
17. A positive integer, if 100 is a perfect square, if 168 is another perfect square, then the positive integer is______ ．
18. What is the difference between the product of 7 / 15 and 5 / 2 minus the quotient of 7 / 58 divided by 29 / 5?
19. Given that a plus B equals three and ab equals one, find the value of a plus a and B of B
20. Let {an} satisfy a1 + 3a2 + 3 & # 178; A3 +... + 3 ^ (n-1) an = n / 3, n ∈ n + * (1) find the general term of {an}; (1) Find the general term of the sequence {an}; (2) let BN = n / an, find the first n term and Sn of the sequence {BN} a1+3a2+3^2a3+······+3^(n-1)an=n/3 a1+3a2+3^2a3+······+3^(n-2)a(n-1)=(n-1)/3 Subtraction of two formulas 3^(n-1)an=n/3-(n-1)/3 3^(n-1)an=(n-n+1)/3 3^(n-1)an=1/3 an=1/3^n bn=n/an =n/(1/3^n) =n*3^n Why? a1+3a2+3^2a3+······+3^(n-1)an=n/3 a1+3a2+3^2a3+······+3^(n-2)a(n-1)=(n-1)/3 Subtraction of two formulas 3^(n-1)an=n/3-(n-1)/3 How do you get this by subtracting 3 ^ (n-1) an = n / 3 - (n-1) / 3 Subtraction is not to get 3 ^ (n-1) an-3 ^ (n-2) a (n-1) = n / 3 - (n-1) / 3. How can it be transformed into 3 ^ (n-1) an = n / 3 - (n-1) / 3