All Questions
16
questions
0
votes
1
answer
37
views
Checking an induction proof for a summation.
In my textbook practice problem, I want to prove the following by induction:
$$\sum_{i=0}^{n}\sum_{k=0}^{n-i}a_{i, k} = \sum_{m=0}^{n}\sum_{k=0}^{m}a_{m-k, k}$$
For my "$n+1$ implies $n$" ...
- 351
0
votes
1
answer
23
views
Evaluation of an indexed sum. [closed]
I am going through a textbook solution to an induction problem and I'm not sure why it says this summation holds.
$\sum_{m=0}^0 \sum_{k=0}^m a_{m - k, k} = a_{0, 0}$
This is very simple I know, but ...
- 351
3
votes
1
answer
66
views
How can i simplify the following formula: $\sum\limits_{i,j=1}^{n}(t_{j}\land t_{i})$?
Consider the following time discretization $t_{0}=0< t_{1} < ... < t_{n} = T$ of $[0,T]$ where the time increments are equal in magnitude, i.e. $t_{j}-t_{j-1}=\delta$.
How can i simplify the ...
- 1,838
2
votes
3
answers
92
views
Proof $\sum_{k=1}^{2n} (-1)^{1+k}\frac{1}{k} = \sum_{k=1}^{n}\frac{1}{n+k}$ by induction. [duplicate]
I'm trying to show the following formula:
$$\sum_{k=1}^{2n} (-1)^{1+k}\frac{1}{k} = \sum_{k=1}^{n}\frac{1}{n+k}$$
I have already verified the formula with $n=1$, now I continue with the induction:
$$ \...
1
vote
2
answers
52
views
Have I followed through on this proof by induction correctly?
We are asked to prove that $1^3+2^3+\cdots+n^3 = (1+2+\cdots + n)^2$ by induction.
Basis: $n=1 \\ 1^3 = 1^2 \quad\checkmark \\ n=2 \\ 1^3+2^3 = (1+2)^2 = 9 \quad\checkmark
$
We rewrite both series ...
- 119
1
vote
3
answers
88
views
Bounded sequence of reals
Suppose that $\{a_{n} \}$ is a sequence of real numbers that satisfy $a_{i} + a_{j} \geq a_{i+j}$.
Then prove $$a_{n} \leq \sum_{i=1}^{n} \frac{a_{i}}{i}$$.
I tried to use straight-up induction, but ...
- 2,096
1
vote
6
answers
143
views
Prove that $\sum_{k=0}^{n}{\frac{1}{k!}}\leq 3$ $\forall n\in \mathbb{N}$
Prove that $\sum_{k=0}^{n}{\frac{1}{k!}}\leq 3$ $\forall n\in \mathbb{N}$
I proved it by induction:
"$n=1$" $ \quad \sum_{k=0}^{1}{\frac{1}{k!}}\leq 3 \quad \checkmark$
"$n \implies n+1$": $$\...
- 2,482
1
vote
1
answer
123
views
Suppose $(a)_{j=1}^{\infty}$ is a sequence of real numbers. prove by induction on n that $|\sum_{j=1}^{n}a_j|\leq\sum_{j=1}^{n}|a_j|$
This is a proof that my teacher gave I'm having a hard time with the last line of the proof.
Suppose $(a)_{j=1}^{\infty}$ is a sequence of real numbers. prove by induction on n that $$|\sum_{j=1}^{n}...
- 513
0
votes
3
answers
91
views
Prove through induction that $\sum_{a=1}^{b}a(b-a)=\frac{b(b-1)(b+6)}{6}$ [closed]
Given that $\sum_{a=1}^{b}a=\frac{a(a+1)}{6}$ prove through induction that $$\sum_{a=1}^{b}a(b-a)=\frac{b(b-1)(b+6)}{6}$$
Normally I would start by showing that this statement is true for $b=1$ and ...
- 1,928
-1
votes
3
answers
112
views
What does $\sum_{k=1}^{n}k=\frac{n(n+1)}{2}$ mean in induction? [closed]
If you're given that
$$\sum_{k=1}^{n}k=\frac{n(n+1)}{2}$$
and let's say that prove that
$\sum_{k=1}^{n}k+\frac{1}{k+n}=blah$, $n>0$ what does $k$ in itself equal? I mean if you had to put $k$ ...
- 1,928
4
votes
2
answers
425
views
How to prove this summation formula?
Mathematica is able to calculate the following sum:
\begin{align}
&\sum_{k,l=0}^{n,m}\frac{(-1)^{n+m-l-k}(2m)!(2n+1)!(2 )^{2k+2l}}{(2k+1)!(2l)!(n-k)!(m-l)!}(k+l)! \nonumber\\
=&\frac{(-1)^{...
- 147
0
votes
3
answers
109
views
Mathematical induction proof problem [closed]
How to prove with mathematical induction that
$$\sum _{k=1}^{n}\frac{2k-1}{2^k}=3-\frac{2n+3}{2^n}$$
if $n \in \mathbb N$?
- 9
4
votes
3
answers
384
views
Induction on inequalities: $\frac1{1^2}+\frac1{2^2}+\frac1{3^2}+\ldots+\frac1{n^2}<2$ [duplicate]
I am trying to solve this inequality by induction. I just started to learn induction this week and all the inequalities we had been solved were like an equation less than another equation (e.g. $n! \...
- 45
2
votes
1
answer
4k
views
The identity $a^n-b^n=(a-b) (\sum_{i=0}^{n-1}a^ib^{n-1-i})$ [duplicate]
How do I use finite induction to prove that
$$a^n-b^n=(a-b) (\sum_{i=0}^{n-1}a^ib^{n-1-i}), \forall a,b\in \Bbb{R}\space \text{and} \space \forall n \in \Bbb{N}?$$
Ok, for $n=2$ it's fine. $a^2-b^2=(a-...
- 2,773
5
votes
3
answers
4k
views
Proof of Equation by Well Ordering Principle
I have an assignment question
Prove by either the Well Ordering Principle or induction that for all nonnegative integers $n$: $$\sum_{k=0}^n k^3 = \left(\frac{n(n+1)}{2}\right)^2.$$
I am able to ...
- 93