Prove by induction: "For all \( n \in \mathbb{N} \), \( \sum_{k=1}^{n} k \cdot k! = (n+1)! - 1 \).

Prove that  ∑_(n=1)^∞ (nn/n!) = ∞ Using induction : (11/1!)=1≥1, (22/2!)=2≥1

Prove if the following sequences are monotonic and find their limit where possible a)(n-1)(n+2)/(n+1)(n-3)b)(2n^2-1)/(2n^2+1)

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. Let $n \in \mathrm{N}^{*}$ and the equation:$$C_{2 n+1}^{1} x^{n}-C_{2 n+1}^{3} x^{n-1}+C_{2 n+1}^{5} x^{n-2}-\ldots+(-1)^{k} C_{2 n+1}^{2 k+1} x^{n-k}+\ldots+(-1)^{n} C_{2 n+1}^{2 n+1}=0$$Show that:a) the equation admits as roots the numbers::$\mathrm{x}_{\mathrm{k}}=\operatorname{ctg}^{2} \frac{k \pi}{2 n+1}$, for every $k \in\{1.2, \ldots . n\}$  b) $\frac{\pi^{2} n(2 n-1)}{3(2 n+1)^{2}}<\sum_{k=1}^{n} \frac{1}{k^{2}}<\frac{2 \pi^{2} n(n+1)}{3(2 n+1)^{2}}$ c) $\frac{\pi^{4} n(2 n-1)\left(4 n^{2}+10 n-9\right)}{45(2 n+1)^{4}}<\sum_{k=1}^{n} \frac{1}{k^{4}}<\frac{8 \pi^{4} n(n+1)\left(n^{2}+n+3\right)}{45(2 n+1)^{4}}$  d) $\lim _{n \rightarrow \infty} \sum_{k=1}^{n} \frac{1}{k^{2}}=\frac{\pi^{2}}{6}$  e) $\lim _{n \rightarrow \infty} \sum_{k=1}^{n} \frac{1}{k^{4}}=\frac{\pi^{4}}{4}$  Please check the attached picture for details. I need complete solution with explanation please. the all fifth subpoints are for the same problem and they are conected between them, but if…

Could you plz prove "upper reimn sum is always greater or to lower reimen sum"thanks

Discrete math

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Use Mathematical induction to show that H sub(2^n) is less than or equal to 1 + n.

These are proofs please answer all questions and parts and show the work neatly. 1. Use generalized induction to prove that n! < n^2 for all integers n ≥ 2. 2. The Fibonacci sequence{fn}= 1,1,2,3,5,8,13,21, . . .is defined recursively byfn+1=fn+fn−1for alln≥2, wheref1=f2= 1. In each case, use complete induction (witha= 1,b= 2) to prove the given statement.(a) Prove thatfn<2nfor all integersn≥1. (b) Prove thatfn=(1 +√5)n−(1−√5)n2n√5for all integersn≥1.