Intereting Posts

Convergence of integrals in $L^p$
Continuity of Integration
What is the probability of of drawing at least 1 queen, 2 kings and 3 aces in a 9 card draw of a standard 52 card deck?
If $a+b=1$ so $a^{4b^2}+b^{4a^2}\leq1$
Question about generating function in an article
Function always continuous in a Sobolev Space?
Embedding ordinals in $\mathbb{Q}$
Ambiguity of Quotient Groups
Visualizing Commutator of Two Vector Fields
How to prove Euler's formula: $e^{it}=\cos t +i\sin t$?
Formal logic systems, how do we prove theorems about them?
Big O notation sum rule
Problem on circles, tangents and triangles
Percentage to absolute value within another range?
Probability that a natural number is a sum of two squares?

I am trying to prove the following by Mathematical Induction:

$$\sum_{i=1}^n i\cdot i! = (n+1)!-1\quad\text{for all integers $n\ge 1$}$$

My proof by Induction follows:

First prove $P(1)$ is true,

- Coloring dots in a circle with no two consecutive dots being the same color
- $x+1/x$ an integer implies $x^n+1/x^n$ an integer
- Mathematical Induction for Recurrence Relation
- Is empty set element of every set if it is subset of every set?
- Can someone explain me this summation?
- In naive set theory ∅ = {∅} = {{∅}}?

$$\sum_{i=1}^1 i\cdot i! = (1+1)! – 1$$

Then, for all integers $k >= 1$, if $P(k)$ is true then $P(k+1)$ is also true,

$$ \sum_{i=1}^k i\cdot i! = (k+1)! – 1$$

Then, we must show that $P(k+1)$ is true

$$ \sum_{i=1}^{k+1} i\cdot i! = ((k+1)+1)! – 1\\

\sum_{i=1}^{k+1} i\cdot i! = (k+2)! – 1$$

I am currently struggling with the next step to show that $P(k)$ is equal to $P(k+1)$. How would I finish this prove by induction?

- Prove that if $m^2 + n^2$ is divisible by $4$, then both $m$ and $n$ are even numbers.
- Use Mathematical Induction to prove that $\frac{1}{1 \cdot 2} + \frac{1}{2 \cdot 3} +…+\frac{1}{n(n+1)}=1-\frac{1}{n+1}$
- Coin flipping probability game ; 7 flips vs 8 flips
- Another counting problem on the number of ways to place $l$ balls in $m$ boxes.
- What is the purpose of implication in discrete mathematics?
- Estimate of n factorial: $n^{\frac{n}{2}} \le n! \le \left(\dfrac{n+1}{2}\right)^{n}$
- Count Exclusive Partitionings of Points in Circle, Closing Double Recurrence?
- Do factorials really grow faster than exponential functions?
- The product of n consecutive integers is divisible by n! (without using the properties of binomial coefficients)
- Proving $(A\cup B)\cap(B\cup C)\cap(C\cup A)=(A\cap B)\cup (A\cap C)\cup (B\cap C)$?

Hint: $$\sum_{i=1}^{k+1} i\cdot i! = (k+1)\cdot(k+1)! + \sum_{i=1}^{k} i\cdot i! $$

by the definition of the $\sum$ operator.

Also, by the induction hypothesis, you know that $\sum_{i=1}^{k} i\cdot i!$, which appears on the right-hand side, is equal to $(k+1)!-1$.

If you want an alternative to induction, use telescoping sums:

$$

\sum_{i=1}^ni\cdot i!=\sum_{i=1}^n(i+1-1)\cdot i!=\sum_{i=1}^n(i+1)\cdot i!-1\cdot i!

$$

$$

=\sum_{i=1}^n (i+1)!-i!=(n+1)!-n!+n!-\ldots-2!+2!-1!=(n+1)!-1.

$$

- Why do we consider Lebesgue spaces for $p$ greater than and equal to $1$ only?
- Vector spaces of the same finite dimension are isomorphic
- Prove the open mapping theorem by using maximum modulus principle
- Why doesn't the equation have a solution in $\mathbb{Q}_2$?
- Schur skew functions
- Is $g(u)= \frac{E }{E }$ decreasing in $u$
- Check my answer: Prove that every open set in $\Bbb R^n$ is a countable union of open intervals.
- Trace minimization when some matrix is unknown
- If every nonzero element of $R$ is either a unit or a zero divisor then $R$ contains only finitely many ideals?
- Find $v_p\left(\binom{ap}{bp}-\binom{a}{b}\right)$, where $p>a>b>1$ and $p$ odd prime.
- Hessian Related convex optimization question
- Conjectured closed form for $\operatorname{Li}_2\!\left(\sqrt{2-\sqrt3}\cdot e^{i\pi/12}\right)$
- recurrence relation number of bacteria
- Two Dimensional Lie Algebra
- Show that there is $\xi$ s.t. $f(\xi)=f\left(\xi+\frac{1}{n}\right)$