0
votes
2answers
37 views

How to use the Comparison Test to investigate the convergence of $\sum (\ln n)/n^\alpha$?

Let $$\sum\limits_{n=1}^\infty \frac{\ln n}{n^\alpha}, \alpha\in\Bbb{R}$$ I need to investigate the convergence of this series. I've read that since the series is positive for all $n$ then it ...
0
votes
2answers
70 views

Mathematical Induction Problem with Fraction

$$(3n-2)^2=\frac{n(6n^2-3n-1)}{2}$$ I can't seem to solve it out to the point where I can prove it right or wrong. I always hit some sort of roadblock where I don't have enough info to prove it ...
0
votes
1answer
31 views

Recursive sequence problem

$$U(n+1) = (6+U(n))^{1/3},\text{ and } U(0) = 1.$$ Prove by induction that for all positive integers $n, U(n)$ is increasing. Prove by induction that for all positive integers $n, U(n) \leq 2$ ...
2
votes
2answers
62 views

Consider the sequence defined recursively by $U(n+1) =\frac{1}{3-U(n)}$ and $U(0) = 2$. [closed]

Prove by induction that for all positive integers $n, U(n)$ is decreasing Prove by induction that for all positive integers $n, U(n) > 0$ (namely, the sequence is bounded from below) Does the ...
3
votes
1answer
78 views

Proving that nth derivate of $x e^{-x}$ is $(-1)^n (e^{-x})(x-n)$ by induction.

I'm quite stuck on this. How would you prove that the $n^{th}$ derivative of $x e^{-x}$ if the $(-1)^n (e^{-x})(x-n)$ by induction? I did: $\frac{d}{dx}(x e^{-x})=(e^{-x}) - x(e^{-x})$ Now I have ...
0
votes
1answer
40 views

Proof by induction valid or not?

Prove by induction the following: $$\sum_{i=0}^n x^i = \frac{1-x^{n+1}}{1-x}$$ We want: $$x^0+x^1+ \ldots + x^n = \frac{1-x^{n+1}}{1-x}$$ I try this for $i=1$ and it works, so I have an initial ...
2
votes
3answers
128 views

Proof that $\binom{ n}{k} \in \mathbb N$

This problem is from Spivak. Give another proof that $\binom{n}{k}$ is a natural number by showing that $\binom{n}{k}$ is the number of sets of exactly $k$ integers each chosen from $1, \ldots,n$. I ...
1
vote
1answer
114 views

Prove $\frac {1} {1+x^2} = \sum^{n-1}_{j=0} (-1)^j x^{2j} + (-1)^n \frac {x^{2n}} {1+x^2}, x\in \mathbb R, n \in \mathbb N$ by induction

Prove $\frac {1} {1+x^2} = \sum^{n-1}_{j=0} (-1)^j x^{2j} + (-1)^n \frac {x^{2n}} {1+x^2}, x\in \mathbb R, n \in \mathbb N$ by induction. $n = 1$: $\sum^{(1)-1}_{j=0} (-1)^j x^{2j} + (-1)^n \frac ...
0
votes
2answers
55 views

Induction - Discrete Mathematics [closed]

Prove by induction on n that for any $n ≥ 2$ , any sequence of non-zero real numbers $\large a_1,a_2,...,a_n$ that starts with a negative number, i.e., $a_1 < 0$ and ends with a positive number, ...
1
vote
1answer
68 views

Stuck on inductive step: $2^x > x^n$ when $x\rightarrow \infty$

I want to show that $2^x > x^n$ when $x \rightarrow \infty$ for all $n \in \mathbb{N}$. I'm trying to do it by induction over $n$. The base case, $n = 1$, is true: $2^x > x$ when $x \rightarrow ...
2
votes
6answers
312 views

If $a_1,a_2,\ldots,a_n>0$ and $a_1+a_2+\ldots+a_n<\frac{1}{2}$, prove that $(1+a_1)(1+a_2)\ldots(1+a_n)<2$.

If $a_1,a_2,\ldots,a_n>0$ and $a_1+a_2+\cdots+a_n<\frac{1}{2}$, prove that $$(1+a_1)(1+a_2)\cdots(1+a_n)<2$$ I've tried Hölder's inequality (the same result can easily be derived using ...
0
votes
1answer
19 views

Conjecture based on limited trail followed by inductive proof

My syllabus says: recognise situations where conjecture based on a limited trail followed by inductive proof is a useful strategy, and carry this out in simple casses e.g. find the nth derivative ...
1
vote
1answer
89 views

Integration using induction question

Assume $f : [0, 1] \to \mathbb{R}$ is continuous and arbitrarily often differentiable on $(0, 1)$ (i.e. $f$ is smooth). Denote by $f^{m}$ the $m\text{-th}$ derivative of $f$ with $m∈\mathbb{N}$ and ...
1
vote
1answer
53 views

Am I understanding induction correctly?

Here is an induction proof that I have written for my homework and I want to know if I am understanding this correctly: Prove that for: $ \sum\limits_{i=0}^n i^2 = \frac{n(n+1)(2n+1)}{6}$ My proof: ...
0
votes
2answers
47 views

Bernoulli's inequality by induction

I'm proving Bernoulli's inequality by induction but I noticed something strange. See wikipedia proof: http://en.wikipedia.org/wiki/Bernoulli's_inequality Notice how they multiply both sides of the ...
0
votes
3answers
99 views

Estimate the factorial $n!$ starting with the integral of $1/x$

This is a 3-part problem concerning an estimate for the factorial $n!$ a. By considering the graph of $y=\frac{1}{x}$ explain why $$\frac{1}{k+1} < \int\limits_{k}^{k+1} \frac{\mathrm ...
2
votes
2answers
69 views

Induction - what happened? Summation

So I have the general summation formula that I was to prove using mathemathical induction on a Calculus level. For all $n = {1,2,3,...}$ we have: $$\sum_{j=n}^{2n-1} (2j + 1) = 3n^2$$ In the ...
1
vote
1answer
337 views

Proof by Induction problem. Unsure of sigma notation. [duplicate]

I want to use proof by induction to verify $$\sum_{j=n}^{2n-1} (2j+1) = 3n^2.$$ First I assume that the expression is valid for some number k, $S(k) = 2k + 1 $ Then k + 1 so that, $S(k+1) = s(k) ...
0
votes
0answers
46 views

Inequality sine power series (induction)

How can we show, for $k\geq 1$ and $x \geq 0$, the inequality below by induction? $\displaystyle \sin x \geq \sum_{n=1}^{2k} (-1)^{n+1} \frac 1{ (2n - 1)! }x^{2n-1} $ The base case $k = 1$ gives ...
2
votes
0answers
77 views

Inequality sine power series

How can we show, for $k\geq 1$ and $x \geq 0$, the inequality below by induction? $\displaystyle \sin x \geq \sum_{n=1}^{2k} (-1)^{n+1} \frac 1{ (2n - 1)! }x^{2n-1} $ The base case $k = 1$ gives ...
1
vote
1answer
28 views

Proving the monotonicity of a recurrence.

Define the following recurrence for $n = 1, 2, \cdots$ $T(n) = ( 1 - \operatorname{H}(\frac{1 - P^{\frac{1}{n}}}{2}))^n$ where $0 < P < 1$ is a constant, function $\operatorname{H}(\cdot)$ is ...
0
votes
1answer
56 views

proof by induction the following sequence formula

let $a_0 = 0, a_1=1,a_n = \frac{a_{n-1} + a_{n-2}}{2} $ prove with induction that $a_n = \frac23 \left(1 + \frac{(-1)^{n+1}}{2^n} \right) $ i assumed $a_k$ was equal to the given formula ...
-1
votes
1answer
60 views

Induction On $a_{n+1}$ Sequence

We define a sequence of rational numbers $\{a_n\}$ by putting $a_1=2$ and $$a_{n+1}=3−\frac1{a_n}$$ for all $n\in\Bbb N$. Put $$\alpha=\frac{3+\sqrt5}2\;.$$ I've shown that ...
2
votes
4answers
120 views

Recurrence sequence limit

I would like to find the limit of $$ \begin{cases} a_1=\dfrac3{4} ,\, & \\ a_{n+1}=a_{n}\dfrac{n^2+2n}{n^2+2n+1}, & n \ge 1 \end{cases} $$ I tried to use this - $\lim ...
1
vote
3answers
80 views

Proof by induction proof by using positive numbers n

I was able to prove by induction, that for $k=1$, $\frac{d}{dx} e^{kx} = ke^{kx}$. Assuming the rule is true for $k=n$, I am supposed to show that the rule is true for $ k = n+1$ I have no idea ...
-1
votes
3answers
112 views

Prove $2^n > 10n^2$ for all sufficiently large integers n.

How do I prove $2^n > 10n^2$ inductively? I know you can prove this to be true using calculus (i.e. taking derivatives). But how would I do it inductively?
3
votes
2answers
125 views

Tallest bubble tower induction proof

A hemispherical bubble is placed on a spherical bubble of radius $1$. A smaller hemispherical bubble is then placed on the first one. This process is continued until $n$ chambers, including the ...
0
votes
2answers
401 views

Induction step for $\sum\limits_{i=1}^n i^2 = \frac{n(n+1)(2n+1)}{6}$

I want to prove by induction that, $\sum\limits_{i=1}^n i^2 = \frac{n(n+1)(2n+1)}{6}$ OK I got the initial step, however, I have problems with the induction step: Here is what I tried: ...
0
votes
2answers
158 views

Strong Induction on Inequalities

I'm asked to indicate which natural numbers $n$ each of the below inequality is true, and then I am required to prove this via induction, but I'm wondering what that means... Strong induction? ...
3
votes
1answer
110 views

A formula for n-derivative of the inverse of a function?

Let $y=f^{-1}(x)$. As we know: \begin{align} \frac{\mathrm{d} y}{\mathrm{d} x}=\frac{1}{{f}'(y)} \end{align} Thereof we have: \begin{align} \frac{\mathrm{d^2} y}{\mathrm{d} ...
2
votes
2answers
337 views

Induction Proof for a series expansion of a function

I have done induction proofs of many different types, but trying to prove by induction that a derivative from the Taylor series expansion of a function has me stumped in terms of how to get the final ...
3
votes
5answers
305 views

What does “Prove by induction” mean?

What does "Prove by induction" mean ? I've heard it a lot! Would you mind giving me an example? Thanks
4
votes
3answers
471 views

induction proof: $\sum_{k=1}^nk^2 = \frac{n(n+1)(2n+1)}{6}$

I encountered the following induction proof on a practice exam for calculus: $$\sum_{k=1}^nk^2 = \frac{n(n+1)(2n+1)}{6}$$ I have to prove this statement with induction. Can anyone please help me ...
2
votes
2answers
115 views

Prove that $n! < n^n $ where n >1 and is an integer , why do some people say my solution is wrong?

Prove that $n! < n^n $ where n >1 and is an integer. Lets skip the base case cause its trivial. Assume that: $$ k! < k^k = $$ Inductive step: $$(k+1)! < (k+1)^{k+1} =$$ $$(k)!(k+1) < ...
0
votes
1answer
81 views

Why is this summation formula wrong?

This is the alternate form of the summation formula: $$ \sum^{n}_{k=0} a(c)^k = \frac{ac^{n+1} - a}{c - 1} $$ so why is this wrong? $$ \sum^{n}_{k=0} (-\frac{1}{2})^k = \frac{(-\frac{1}{2})^{n+1} - ...
0
votes
0answers
91 views

Improper integral $\int_{0}^{\infty}\frac{x^n}{x^{m+n+1}} \ dx=\frac{n! {(m-1)}!}{(m+n)!}.$

How can I prove that $$\int_{0}^{\infty}\frac{x^n}{x^{m+n+1}} \ dx=\frac{n! {(m-1)}!}{(m+n)!}\quad ?$$ I tried to do induction on $n$ and on $m$, separately, but I could only do the base case ($n=1$ ...
0
votes
1answer
205 views

Prove by Induction using Baseline and splitting into LHS & RHS?

I'm having trouble with this equation mainly because it has a couple of odd things with it, and its these that have thrown me off as i'm not to sure how to tackle them. The equation is: ...
0
votes
1answer
47 views

Check Logic for Induction Of T(n) Equation

I've just finished an induction equation, however i'm a little hit and miss about whether its actually right on not. Mainly in my working out, I'd really appreciate if you could have a look at it and ...
2
votes
1answer
139 views

Convergence of a power series function

Consider the following differential equation: $$w''(x)+p(x)w'(x)+q(x)w(x)=r(x)$$ with the initial condition of $w(0)=w_0,\ w'(0)=w_1$, and $$w_{n+2}=\frac{r_{n+2}-(n+1)p_0w_{n+1}-\sum_{k=0}^n w_k ...
2
votes
5answers
76 views

Proof for power functions

Which is greater? $\sqrt{n}^{\sqrt{n+1}}$ or $\sqrt{n+1}^\sqrt{n}$ I know that $\sqrt{n}^{\sqrt{n+1}}$ is greater but I tried using induction and I couldn't figure it out. Thanks for the help.
5
votes
4answers
170 views

Proving ${(n+2)!}< \left(n(\sqrt{2}-1)+\sqrt{2}\right)^{n+2}$

While simplifying an inequality, this inequality was derived: $${(n+2)!}< \left(n(\sqrt{2}-1)+\sqrt{2}\right)^{n+2},\quad\quad\quad\quad n\in \mathbb{N}$$ Do you have any idea to prove it? It is ...
2
votes
1answer
228 views

How do I implement an induction argument to show that we may assume that a partition $Q$ of $[a,b]\in{R}$ which refines $P$ has only one more point?

There is a lemma (32.2) in Kenneth A. Ross's Elementary Analysis text, that states: Let $f$ be a bounded function on $[a,b]$. If $P$ and $Q$ are partitions of $[a,b]$ and $P\subseteq{Q}$, then ...
1
vote
3answers
80 views

Induction prove, how to come to $n\cdot(n+1)$

I am trying to solve an induction problem. Here are the steps for the example. Prove this equation $$ 1\cdot2 + 2\cdot3 + 3\cdot 4 + 4\cdot 5+\dots + \cdots +(n-1)\cdot n ...
1
vote
1answer
83 views

Bernoulli polynomials, Apostol

Define Bernoulli polynomials as: $P_0(x)=1$, $P'_n(x)=nP_{n-1}(x)$, $\int_0^1P_n(x)=0$ if $n\geq1$ Need to prove that for $n\geq2$ we have $$\sum_{r=1}^{k-1} r^n= \frac{P_{n+1}(k)-P_{n+1}(0)}{n+1}$$ ...
1
vote
1answer
268 views

Calculus the n'th derivative of $y_{n}=x^{n-1}e^{1/x}$

Without using Mathematical Induction to calculus the n'th derivative of the following function. $y_{n}=x^{n-1}e^{1/x}$ , $n\in\mathbb{N}$ Find : $\frac{d^n}{dx^n}y_n$ I tried to finish the question ...
1
vote
2answers
136 views

Higher order differentiation of logarithm function by induction

Let $f(x)=\log(x)$ (the natural logarithm). I'm asked to find a system in $f', f'', f^{(3)}, f^{(4)}$ and use induction to prove my system is correct. Edit: After the comments I now have the system ...
3
votes
1answer
216 views

I want inductively prove that $f^{(n)}(0)=0$ for all n.

Define a function $f(x)$ such that: \begin{cases} \exp(-1/x^2), & \text{if } x>0, \\ 0, & \text{if } x\leqslant 0. \\ \end{cases} I want to inductively prove that $f^{(n)}(0)=0$ for all ...
5
votes
2answers
473 views

Is this the correct way to perform mathematical induction? (re: derivative of $z^n$ is $nz^{n-1}$)

Here's a question: Derive $$\frac{\mathrm{d}}{\mathrm{d}z} z^n = nz^{n-1},$$ when n is a positive integer by using mathematical induction and and the derivative of a product of two functions ...
5
votes
2answers
797 views

Proof of Product Rule for Derivatives using Proof by Induction

I am trying to understand the proof of the General Result for the Product Rule for Derivatives by reading this. Relevant parts are as follows: Basis for the induction $$ D_x \left({f_1 ...
0
votes
2answers
257 views

Induction and Integral Question (sum of 5th powers, integral of $x^5$)

I am quite lost on this question: (a) For $n\in \mathbb{N}$, use induction to show that $$\sum_{k=1}^{n}k^{5}=\frac{2n^{6}+6n^{5}+5n^4-n^2}{12}$$ (b) Fix $b>0$. Use the definition of ...