Why is the current I equal and the voltage u always = U1 + U2 +... UN when the circuit is in series

U total = IR total = IX (R1 + R2 +... RN)
U total = IR total = IX (R1 + R2 +... RN) = IR1 + IR2 +... + IRN = U1 + U2 +... UN

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1. Given U1 = 1, U2 = 2, find UN = 2U (n-2) + U (n-1) + 1 Solve how to deduce the general term formula
2. Given the partial sum of series Sn = 2n / N + 1, find U1, U2, UN
3. 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
4. When U1 > 4, UN + 1 = 3un / 4 + 4 / UN, n →∞, UN → x, find X
5. Excuse me: some problems of u = U1 + U2 (string) u = U1 = U2 (Union) Excuse me: u = U1 + U2 (string) u = U1 = U2 (parallel) I = I1 = I2 (string) I = I1 + I2 (parallel) Are the laws of these junior high school experiments only applicable to pure resistance circuits? Not a pure resistance circuit? More detailed points lead to deeper knowledge.
6. If the series ∑ (n = 1) UN converges and the series ∑ (n = 1) VN diverges, try to prove that the series ∑ (n = 1) (UN + VN) diverges and find the detailed solution. Thank you
7. The convergence of series UN ^ 2 is proved
8. Let the series un-un-1 converge and the series VN converge, and prove the absolute convergence of the series UN VN
9. Let the positive series ∑ UN and ∑ VN converge. It is proved that ∑ (UN + VN) ^ 2 also converges ……
10. Let LIM (n →∞) na_ N exists and the series ∑ (n = 1 →∞) n (a)_ n-a_ It is proved that the series ∑ (n = 1 →∞) a is convergent_ N convergence
11. If ∑ UN converges, does ∑ u2n converge? Conversely, does ∑ u2n converge and ∑ UN converge?
12. It is proved that if the series ∑ UN satisfies (1) Limun = 0, (2) ∑ (u2n-1 + u2n) convergence, then ∑ UN converges
13. A series ∑ UN converges. How can we prove that its odd term ∑ u2n-1 also converges?
14. If the series UN converges to s, then the series (UN + UN + 1) converges to s
15. If the series ∑ UN converges to s, the series ∑ [UN + UN + 1] converges to s {n from 1 to infinity}
16. 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
17. Let the positive series ∑ UN converge and the sequence {VN} be bounded. It is proved that the series ∑ UN} is absolutely convergent
18. Let the series ∑ (n = 1) UN converge and ∑ UN = u, then the series ∑ (UN + U (n + 1)) =?
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