Another way to record/save videos from Periscope (Download now!)
If you live in the US, (or know how to download apps from the US App Store) you certainly download the following:
Its a new app called Vidyo, it allows you to record your iPhone screen without jailbreaking your phone.
You can read more details in the following page: https://www.macstories.net/ios/vidyo-a-screen-recorder-for-ios/
PS: DOWNLOAD IT NOW BEFORE APPLE PULLS IT FROM THE APP STORE