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Bug introduced after 5.2, fixed in 12.2 or earlier.

The function

  Pochhammer[1 + n, n]

tends to infinity. We have

  FunctionExpand[Pochhammer[1 + n, n]]
  (* (2^(2*n)*Gamma[1/2 + n])/Sqrt[Pi] *)

But following result

  Series[Pochhammer[1 + n, n] , {n, Infinity, 1}]
  (* 1 + O[1/n]^2 *)

is a bug! Tested under versions 10.2 and 7.0, same results. Interesting is that

  Limit[Pochhammer[1 + n, n]/(2^(2*n + 1/2)*n^n/E^n), n -> Infinity]

is correct.

Bug introduced after 5.2, fixed between 12.0.0 and 12.1.1.

The function

  Pochhammer[1 + n, n]

tends to infinity. We have

  FunctionExpand[Pochhammer[1 + n, n]]
  (* (2^(2*n)*Gamma[1/2 + n])/Sqrt[Pi] *)

But following result

  Series[Pochhammer[1 + n, n] , {n, Infinity, 1}]
  (* 1 + O[1/n]^2 *)

is a bug! Tested under versions 10.2 and 7.0, same results. Interesting is that

  Limit[Pochhammer[1 + n, n]/(2^(2*n + 1/2)*n^n/E^n), n -> Infinity]

is correct.

The function

  Pochhammer[1 + n, n]

tends to infinity. We have

  FunctionExpand[Pochhammer[1 + n, n]]
  (* (2^(2*n)*Gamma[1/2 + n])/Sqrt[Pi] *)

But following result

  Series[Pochhammer[1 + n, n] , {n, Infinity, 1}]
  (* 1 + O[1/n]^2 *)

is a bug! Tested under versions 10.2 and 7.0, same results. Interesting is that

  Limit[Pochhammer[1 + n, n]/(2^(2*n + 1/2)*n^n/E^n), n -> Infinity]

is correct.

Bug introduced after 5.2, fixed in 12.2 or earlier.

The function

  Pochhammer[1 + n, n]

tends to infinity. We have

  FunctionExpand[Pochhammer[1 + n, n]]
  (* (2^(2*n)*Gamma[1/2 + n])/Sqrt[Pi] *)

But following result

  Series[Pochhammer[1 + n, n] , {n, Infinity, 1}]
  (* 1 + O[1/n]^2 *)

is a bug! Tested under versions 10.2 and 7.0, same results. Interesting is that

  Limit[Pochhammer[1 + n, n]/(2^(2*n + 1/2)*n^n/E^n), n -> Infinity]

is correct.