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  • Sum of series:
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  • ln((n+2)/n) ln((n+2)/n)
  • 2(x-6)^(4n)/(n(16)^n)
  • ((-1)^(n-1))*(5^n+2)*x^n
  • Identical expressions

  • factorial(k+ three)*(three / twenty)^k*e^(t*k)/((six *factorial(k)))
  • factorial(k plus 3) multiply by (3 divide by 20) to the power of k multiply by e to the power of (t multiply by k) divide by ((6 multiply by factorial(k)))
  • factorial(k plus three) multiply by (three divide by twenty) to the power of k multiply by e to the power of (t multiply by k) divide by ((six multiply by factorial(k)))
  • factorial(k+3)*(3/20)k*e(t*k)/((6*factorial(k)))
  • factorialk+3*3/20k*et*k/6*factorialk
  • factorial(k+3)(3/20)^ke^(tk)/((6factorial(k)))
  • factorial(k+3)(3/20)ke(tk)/((6factorial(k)))
  • factorialk+33/20ketk/6factorialk
  • factorialk+33/20^ke^tk/6factorialk
  • factorial(k+3)*(3 divide by 20)^k*e^(t*k) divide by ((6*factorial(k)))
  • Similar expressions

  • factorial(k-3)*(3/20)^k*e^(t*k)/((6*factorial(k)))

Sum of series factorial(k+3)*(3/20)^k*e^(t*k)/((6*factorial(k)))



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The solution

You have entered [src]
  oo                     
____                     
\   `                    
 \                 k  t*k
  \   (k + 3)!*3/20 *E   
  /   -------------------
 /            6*k!       
/___,                    
k = 1                    
$$\sum_{k=1}^{\infty} \frac{e^{k t} \left(\frac{3}{20}\right)^{k} \left(k + 3\right)!}{6 k!}$$
Sum(((factorial(k + 3)*(3/20)^k)*E^(t*k))/((6*factorial(k))), (k, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{e^{k t} \left(\frac{3}{20}\right)^{k} \left(k + 3\right)!}{6 k!}$$
It is a series of species
$$a_{k} \left(c t - t_{0}\right)^{d k}$$
- power series.
The radius of convergence of a power series can be calculated by the formula:
$$R^{d} = \frac{t_{0} + \lim_{k \to \infty} \left|{\frac{a_{k}}{a_{k + 1}}}\right|}{c}$$
In this case
$$a_{k} = \frac{e^{k t} \left(k + 3\right)!}{6 k!}$$
and
$$t_{0} = - \frac{3}{20}$$
,
$$d = 1$$
,
$$c = 0$$
then
False

Let's take the limit
we find
$$R = \tilde{\infty} \left(- \frac{3}{20} + e^{- \operatorname{re}{\left(t\right)}}\right)$$
The answer [src]
  oo                     
____                     
\   `                    
 \        k           k*t
  \   3/20 *(3 + k)!*e   
  /   -------------------
 /            6*k!       
/___,                    
k = 1                    
$$\sum_{k=1}^{\infty} \frac{\left(\frac{3}{20}\right)^{k} e^{k t} \left(k + 3\right)!}{6 k!}$$
Sum((3/20)^k*factorial(3 + k)*exp(k*t)/(6*factorial(k)), (k, 1, oo))

    Examples of finding the sum of a series