The 20-year-old had been playing on loan at non-leagues side Eastbourne Borough and Welling United a mere eight months before his transfer to the Gunners from Charlton Athletic.
But Jenkinson believes that he has adapted well to top-flight football and is more than happy with his first season at the club.
"I didn't start how I would have liked to, but an honest assessment would be that as I have progressed, got more minutes under my belt and trained with the boys I definitely feel like I have improved a lot," Jenkinson told Arsenal's official website.
"It was a very steep learning curve for me. I believed I was capable of playing at the highest level, and sometimes it is about getting seen by the right people at the right time.
"I have had some tough times but I feel like I have come out of it a bigger and better person, and a better player.
"In a football career you are going to have ups and downs and tough times, but people have told me it is about how you respond to those tough times that will make or break you as a footballer."
Jenkinson has also faced the tribulations of Premier League football following his sending off at Old Trafford in August – on only his second league appearance in an Arsenal shirt.
"It was all a little bit of a blur," he said. "It was a tough day for me and the club and it was very hard to swallow.
"We had played Udinese in midweek and I felt I had done quite well. That was a massive high, so to go from that to Old Trafford was strange. It happened so quickly.
"It was a disappointing day for me and it took me a while to get over. It was my first taste of a tough result, but things like that make you stronger, and build you as a man and a character."
Overall though, the 20-year-old is happy with his first season as an Arsenal player, citing victories against Chelsea, Spurs and Udinese as his favourite memories of his debut campaign.
"It has been a real rollercoaster," he said. "I have had some tough times and some fantastic times. Chelsea, Spurs and Udinese are games I will never forget.
"It is my first season as an Arsenal player and games like the one at Stamford Bridge were just fantastic. You don't score five goals there very often so that will stay in the memory for a long time.
"I am relatively pleased. I know I have an awful lot to build on, and a lot to improve on. But I have got more in myself to show people, so that is my aim."
Jenkinson moved from Charlton to Arsenal last summer for an undisclosed fee which was believed to be in the region of £1 million.
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