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factorial(m)*0.46^(-10)*((0.46-1)/0.46)^(m-10)*5^m/(((factorial(10)*factorial(m-10)))*(1+5)^(m+1))
Suma de la serie/ factorial(m)*0.46^(-10)*((0.46-1)/0.46)^(m-10)*5^m/(((factorial(10)*factorial(m-10)))*(1+5)^(m+1))

Suma de la serie factorial(m)*0.46^(-10)*((0.46-1)/0.46)^(m-10)*5^m/(((factorial(10)*factorial(m-10)))*(1+5)^(m+1))


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Solución

Ha introducido [src] 
    oo                            
________                          
\       `                         
 \                       m - 10   
  \              /23    \         
   \             |-- - 1|         
    \       m!   |50    |        m
     \    ------*|------|      *5 
      \       10 | /23\ |         
      /   /23\   | |--| |         
     /    |--|   \ \50/ /         
    /     \50/                    
   /      ------------------------
  /                        m + 1  
 /          10!*(m - 10)!*6       
/_______,                         
  m = 10
∑m=10∞5mm!4142651121364997656250000000000(−1+23502350)m−106m+1⋅10!(m−10)!\sum_{m=10}^{\infty} \frac{5^{m} \frac{m!}{\frac{41426511213649}{97656250000000000}} \left(\frac{-1 + \frac{23}{50}}{\frac{23}{50}}\right)^{m - 10}}{6^{m + 1} \cdot 10! \left(m - 10\right)!} 
Sum((((factorial(m)/(23/50)^10)*((23/50 - 1)/(23/50))^(m - 10))*5^m)/(((factorial(10)*factorial(m - 10))*6^(m + 1))), (m, 10, oo))
Radio de convergencia de la serie de potencias
Se da una serie:
5mm!4142651121364997656250000000000(−1+23502350)m−106m+1⋅10!(m−10)!\frac{5^{m} \frac{m!}{\frac{41426511213649}{97656250000000000}} \left(\frac{-1 + \frac{23}{50}}{\frac{23}{50}}\right)^{m - 10}}{6^{m + 1} \cdot 10! \left(m - 10\right)!}
Es la serie del tipo
am(cx−x0)dma_{m} \left(c x - x_{0}\right)^{d m}
- serie de potencias.
El radio de convergencia de la serie de potencias puede calcularse por la fórmula:
Rd=x0+lim⁡m→∞∣amam+1∣cR^{d} = \frac{x_{0} + \lim_{m \to \infty} \left|{\frac{a_{m}}{a_{m + 1}}}\right|}{c}
En nuestro caso
am=15258789062500(−2723)m−10⋅6−m−1m!23488831858138983(m−10)!a_{m} = \frac{15258789062500 \left(- \frac{27}{23}\right)^{m - 10} \cdot 6^{- m - 1} m!}{23488831858138983 \left(m - 10\right)!}
y
x0=−5x_{0} = -5
,
d=1d = 1
,
c=0c = 0
entonces
R=∞~(−5+lim⁡m→∞((2723)9−m(2723)m−10⋅6−m−1⋅6m+2∣m!(m−9)!(m−10)!(m+1)!∣))R = \tilde{\infty} \left(-5 + \lim_{m \to \infty}\left(\left(\frac{27}{23}\right)^{9 - m} \left(\frac{27}{23}\right)^{m - 10} \cdot 6^{- m - 1} \cdot 6^{m + 2} \left|{\frac{m! \left(m - 9\right)!}{\left(m - 10\right)! \left(m + 1\right)!}}\right|\right)\right)
Tomamos como el límite
hallamos
R1=∞~(−5+lim⁡m→∞((2723)9−m(2723)m−10⋅6−m−1⋅6m+2∣m!(m−9)!(m−10)!(m+1)!∣))R^{1} = \tilde{\infty} \left(-5 + \lim_{m \to \infty}\left(\left(\frac{27}{23}\right)^{9 - m} \left(\frac{27}{23}\right)^{m - 10} \cdot 6^{- m - 1} \cdot 6^{m + 2} \left|{\frac{m! \left(m - 9\right)!}{\left(m - 10\right)! \left(m + 1\right)!}}\right|\right)\right)
R=∞~(−5+lim⁡m→∞((2723)9−m(2723)m−10⋅6−m−1⋅6m+2∣m!(m−9)!(m−10)!(m+1)!∣))R = \tilde{\infty} \left(-5 + \lim_{m \to \infty}\left(\left(\frac{27}{23}\right)^{9 - m} \left(\frac{27}{23}\right)^{m - 10} \cdot 6^{- m - 1} \cdot 6^{m + 2} \left|{\frac{m! \left(m - 9\right)!}{\left(m - 10\right)! \left(m + 1\right)!}}\right|\right)\right)
Velocidad de la convergencia de la serie
10.016.010.511.011.512.012.513.013.514.014.515.015.5-500000500000
Respuesta [src] 
 21934509277343750000000000
---------------------------
627753463394042268515136177
21934509277343750000000000627753463394042268515136177\frac{21934509277343750000000000}{627753463394042268515136177} 
21934509277343750000000000/627753463394042268515136177
Gráfico
Suma de la serie factorial(m)*0.46^(-10)*((0.46-1)/0.46)^(m-10)*5^m/(((factorial(10)*factorial(m-10)))*(1+5)^(m+1))

    Ejemplos de hallazgo de la suma de la serie

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