¿Cuál es el valor del límite dado? (Ver detalles)

[math] \ displaystyle \ lim_ {n \ rightarrow \ infty} \ left [\ frac {\ sqrt {n}} {(3 + 4 \ sqrt {n}) ^ {2}} + \ frac {\ sqrt {n }} {\ sqrt {2} (3 \ sqrt {2} +4 \ sqrt {n}) ^ {2}} + \ frac {\ sqrt {n}} {\ sqrt {3} (3 \ sqrt {3 } +4 \ sqrt {n}) ^ {2}} +… + \ frac {1} {49n} \ right] [/ math]

[math] \ displaystyle \ Rightarrow \ lim_ {n \ rightarrow \ infty} \ left [\ frac {\ sqrt {n}} {(3 \ sqrt {1} +4 \ sqrt {n}) ^ {2}} + \ frac {\ sqrt {n}} {\ sqrt {2} (3 \ sqrt {2} +4 \ sqrt {n}) ^ {2}} +… + \ frac {\ sqrt {n}} {\ sqrt {n} (3 \ sqrt {n} +4 \ sqrt {n}) ^ {2}} \ right] [/ math]

[math] \ displaystyle \ Rightarrow \ lim_ {n \ rightarrow \ infty} \ sum_ {r = 1} ^ {n} \ frac {\ sqrt {n}} {\ sqrt {r} (3 \ sqrt {r} + 4 \ sqrt {n}) ^ {2}} [/ math]

[math] \ displaystyle \ Rightarrow \ lim_ {n \ rightarrow \ infty} \ sum_ {r = 1} ^ {n} \ frac {1} {\ sqrt {\ frac {r} {n}} (3 \ sqrt { \ frac {r} {n}} + 4) ^ {2}}. \ frac {1} {n} [/ math]

Ponga, [math] \ displaystyle \ frac {r} {n} = x [/ math]

[math] \ displaystyle \ Rightarrow \ int_ {0} ^ {1} \ frac {dx} {\ sqrt {x} (3 \ sqrt {x} +4) ^ {2}} [/ math]

Ponga, [math] \ displaystyle 3 \ sqrt {x} + 4 = t [/ math]

[math] \ displaystyle \ Rightarrow \ frac {2} {3} \ int_ {4} ^ {7} \ frac {dt} {t ^ {2}} = \ frac {2} {3} \ left [\ frac {-1} {t} \ right] _ {4} ^ {7} = \ frac {2} {3} \ left [\ frac {1} {4} – \ frac {1} {7} \ right] = \ frac {1} {14} [/ math]

Shukriya 🙂