Question 1209826
Let's find a closed form for the sum S_k.

**1. Analyze the Terms**

The general term of the sum is:

T_n = n * n! * (n + 1)

**2. Manipulate the Terms**

We can rewrite the term as:

T_n = n * n! * (n + 1) = n * (n + 1)!

Now, we can rewrite n as (n + 2 - 2):

T_n = (n + 2 - 2) * (n + 1)!

Distribute:

T_n = (n + 2) * (n + 1)! - 2 * (n + 1)!

T_n = (n + 2)! - 2 * (n + 1)!

**3. Apply the Summation**

S_k = Σ T_n (from n = 1 to k)

S_k = Σ [(n + 2)! - 2 * (n + 1)!] (from n = 1 to k)

S_k = [3! - 2 * 2!] + [4! - 2 * 3!] + [5! - 2 * 4!] + ... + [(k + 2)! - 2 * (k + 1)!]

**4. Observe the Pattern**

Notice that we can rearrange the terms:

S_k = [3! + 4! + 5! + ... + (k + 2)!] - 2 * [2! + 3! + 4! + ... + (k + 1)!]

Let's look at the first few terms:

* 3! - 2 * 2! = 6 - 4 = 2
* 4! - 2 * 3! = 24 - 12 = 12
* 5! - 2 * 4! = 120 - 48 = 72

**5. Simplify the Sum**

We can rewrite T_n as:

T_n = (n + 2)! - 2 * (n + 1)! = [(n + 2) * (n + 1)!] - 2 * (n + 1)! = (n + 2 - 2) * (n + 1)! = n * (n + 1)!

Let's try a different approach.

T_n = n * n! * (n + 1) = n * (n + 1)!

Now, we can write:

(n + 1)! = (n + 1) * n!

T_n = n * (n + 1) * n!

We can also write:

(n + 1) * n! = (n + 1)!

T_n = n * (n + 1)!

We can rewrite n as (n + 2 - 2):

T_n = (n + 2 - 2) * (n + 1)! = (n + 2)! - 2(n + 1)!

Now, let's look at the partial sums:

* S_1 = 1 * 1! * 2 = 2
* S_2 = 1 * 1! * 2 + 2 * 2! * 3 = 2 + 12 = 14
* S_3 = 14 + 3 * 3! * 4 = 14 + 72 = 86

Let's try a different manipulation:

T_n = n * n! * (n + 1) = [(n + 2) - 2] * n! * (n + 1) = (n + 2)(n + 1)n! - 2(n + 1)n! = (n + 2)! - 2(n + 1)!

Now, let's sum:

S_k = Σ [(n + 2)! - 2(n + 1)!]

S_k = (3! - 2 * 2!) + (4! - 2 * 3!) + (5! - 2 * 4!) + ... + ((k + 2)! - 2(k + 1)!)

S_k = (k + 2)! - 2(k + 1)! - (2 * 2! - 3!) = (k+2)! - 2(k+1)! - (4 - 6) = (k+2)! - 2(k+1)! + 2

S_k = (k + 2)(k + 1)! - 2(k + 1)! + 2 = (k + 2 - 2)(k + 1)! + 2 = k(k + 1)! + 2

**6. Verify the Closed Form**

* S_1 = 1(2!) + 2 = 2 + 2 = 4 (incorrect)
* S_2 = 2(3!) + 2 = 12 + 2 = 14 (correct)
* S_3 = 3(4!) + 2 = 72 + 2 = 74 (incorrect)

The given closed form is not correct. We must find another form.

T_n = n * n! * (n + 1) = n * (n + 1)! = (n + 1 - 1)(n + 1)! = (n + 1)(n + 1)! - (n + 1)! = (n + 2 - 1)(n+1)!-(n+1)!

T_n = (n+2)!/(n+2) * (n+1) - (n+1)! = (n+2-1)(n+1)!-(n+1)! = (n+2)!(1-1/(n+2)) - (n+1)!

T_n = (n+2)! - (n+1)!-(n+1)! = (n+2)!-(n+1)!-(n+1)!

T_n = (n+2)! - 2(n+1)!

S_k = Σ [(n+2)!-2(n+1)!] = (k+2)! - 2(k+1)! - (3! - 2(2!)) = (k+2)! - 2(k+1)! - (6-4) = (k+2)! - 2(k+1)! - 2

S_k = (k+2)(k+1)! - 2(k+1)! - 2 = (k+2-2)(k+1)! - 2 = k(k+1)! - 2

**Final Answer:** S_k = k(k + 1)! - 2