Clear[λ]

j=0

j=0

coeff=1

Clear[p,k]

p=(λ/k);

coeff (p^j)(((1-p))^(k-j))

Limit[coeff (p^j)(((1-p))^(k-j)),k->∞]

j=1

j=1

coeff=k

Clear[p,k]

p=(λ/k);

coeff (p^j)(((1-p))^(k-j))

Limit[coeff (p^j)(((1-p))^(k-j)),k->∞]

j=2

j=2

coeff=2k(k-1)

Clear[p,k]

p=(λ/k);

coeff (p^j)(((1-p))^(k-j))

Limit[coeff (p^j)(((1-p))^(k-j)),k->∞]

j=3

j=3

coeff=6k(k-1)(k-2)

Clear[p,k]

p=(λ/k);

coeff (p^j)(((1-p))^(k-j))

Limit[coeff (p^j)(((1-p))^(k-j)),k->∞]